Improve 60-pitch "Rube Goldberg" to make it 48 pitches? (what to "temper out"?)

I've been tinkering with this 60-pitch 13-limit scale, and I'm really
pretty happy with it.
But I can't help wondering whether I could get it down to 48 pitches and
be even happier.
There are several features of the scale as it currently stands that I
want to preserve:
1) It has 8:9:10:11:12:13:14:15:16 pretty durn close on all the
corresponding "undertones"--that is, 1/1, 16/15, 8/7, 13/8, 4/3, 16/11,
8/5, 16/9 (with small adjustments)
2) It has adequate chains of "5ths" and "4ths". They don't make a
circle, and that won't bother me for my purposes, which are of course
nefarious.
Chain of "5ths" 1/1, 3/2, 9/8, 22/13, 14/11, 40/21, 10/7, 15/14, 8/5,
6/5, 9/5, 27/20 (with small adjustments)
Chain of "4ths" 1/1, 4/3, 16/9, 13/11, 11/7, 21/20, 7/5, 15/8, 5/4, 5/3,
10/9, 22/15 (with smallish adjustments--the last 22/15 is raised
considerably)
If you look at the Scala file below, you'll see that these have often
been adjusted to make the 4ths and 5ths adequate, if inconsistent, and
neither chain leads back to 1/1-- which is ok for my purposes!
There's a little subtle beating in some of the 8-16ot combinations, but
not so much that they don't sound like themselves. This, and the
"adequate 5ths" are what I want to preserve.
I want to end up with 48 pitches.
Somehow, 12 pitches need to be eliminated from the current 48, but with
the basic idea of 1 and 2 preserved: Slightly wide "5ths", slightly
narrow "4ths", 32/27 is equivalent to 13/11.
I'm going to try to figure this one out myself, but maybe someone else
already has.
Maybe someone would enjoy trying.
(Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which
has patterns of 24,8,24,8,24 mostly--except when it doesn't. There has
to be a logical reason for this scale--Mr. Gene Ward Smith is too
intelligent for it to be frivolous, so clearly I'm not getting it, yet.)

! caleb60.scl60 note 13 Rube Goldberg60!cents PC Approximate ratio
! 0 1/1 53.2 1 33/32 84.5 2 [4/3 below 7/5, or
21/20]104.955 3 17/16111.7 4 16/15 119.44 5 15/14 !128.3 6
14/13138.6 7 13/12150.6 8 12/11165 9 11/10179.1 10 10/9 was
182.4207.2 11 9/8 wide with 3/2!231.2 0 8/7 247.74 1 15/13
265.2 2 7/6 lowered for low 4/3289.2 3 13/11 and tempered 32/27313.6
4 6/5 was orginally 315.6344.1 5 11/9 tempered!359.47 6 16/13
385 7 5/4 400 8414.5 9 14/11 low to go with 22/13435.1 10
9/7454.2 11 13/10!470.781 0 21/16 496.4 1 4/3 low 519.551 2
27/20 536.95 3 15/11551.3 4 11/8563.4 5 18/13!582.5 6 7/5 593.5
7 5/4 above tempered 9 @ 207.2617.5 8 10/7636.6 9 13/9648.7 10
16/11 672 11 22/15 adjusted high to 4/3 over 10/9!703.6 0 3/2
wide729.208 1 32/21 745.8 2 20/13764.9 3 14/9772.6 4
25/16782.5 5 11/7!819 6 8/5 high, originally 813.78 840.53 7
13/8852.6 8 18/11882.7 9 5/3 lowered with 4/3900 10910.789 11
22/13 !933.1 0 12/7952.25 1 26/15 968.8 2 7/4992.8 3 16/9
low with 4/3 1013.6 4 9/51035 5 20/11!1049.4 6 11/61061.4 7
24/131071.7 8 13/71085 9 15/81115.5 10 [4/3 above 10/7, or
40/21]1146.727 11 64/33 !1200 0 2/1

Without tempering, just by removing 12 of the least-important of 60 to make 48:

48-pitch 13 Rube Goldberg
48
!cents PC Approximate ratio
! 0 1/1
53.2 1 33/32
84.5 2 [4/3 below 7/5, or 21/20]
111.7 3 16/15
135 4 13/12 was 138.6
150.6 5 12/11
!
165 6 11/10
179.1 7 10/9 was 182.4
207.2 8 9/8 wide with 3/2
231.2 9 8/7
247.74 10 15/13
265.2 11 7/6 lowered for low 4/3
!
289.2 0 13/11 and tempered 32/27
313.6 1 6/5 was orginally 315.6
344.1 2 11/9 tempered
359.47 3 16/13
385 4 5/4
414.5 5 14/11 low to go with 22/13
!
435.1 6 9/7
454.2 7 13/10
496.4 8 4/3 low
551.3 9 11/8
563.4 10 18/13
582.5 11 7/5
!
617.5 0 10/7
636.6 1 13/9
648.7 2 16/11
672 3 22/15 adjusted high to 4/3 over 10/9
703.6 4 3/2 wide
745.8 5 20/13
!
764.9 6 14/9
782.5 7 11/7
819 8 8/5 high, originally 813.78
840.53 9 13/8
852.6 10 18/11
882.7 11 5/3 lowered with 4/3
!
910.789 0 22/13
933.1 1 12/7
952.25 2 26/15
968.8 3 7/4
992.8 4 16/9 low with 4/3
1013.6 5 9/5
!
1035 6 20/11
1049.4 7 11/6
1061.4 8 24/13
1071.7 9 13/7
1085 10 15/8
1115.5 11 [4/3 above 10/7, or 40/21]
!
1200 0 2/1
On Aug 23, 2010, at 11:29 AM, calebmrgn wrote:

> I've been tinkering with this 60-pitch 13-limit scale, and I'm really pretty happy with it.
>
>
> But I can't help wondering whether I could get it down to 48 pitches and be even happier.
>
> There are several features of the scale as it currently stands that I want to preserve:
>
> 1) It has 8:9:10:11:12:13:14:15:16 pretty durn close on all the corresponding "undertones"--that is, 1/1, 16/15, 8/7, 13/8, 4/3, 16/11, 8/5, 16/9 (with small adjustments)
>
> 2) It has adequate chains of "5ths" and "4ths". They don't make a circle, and that won't bother me for my purposes, which are of course nefarious.
>
> Chain of "5ths" 1/1, 3/2, 9/8, 22/13, 14/11, 40/21, 10/7, 15/14, 8/5, 6/5, 9/5, 27/20 (with small adjustments)
>
> Chain of "4ths" 1/1, 4/3, 16/9, 13/11, 11/7, 21/20, 7/5, 15/8, 5/4, 5/3, 10/9, 22/15 (with smallish adjustments--the last 22/15 is raised considerably)
>
> If you look at the Scala file below, you'll see that these have often been adjusted to make the 4ths and 5ths adequate, if inconsistent, and neither chain leads back to 1/1-- which is ok for my purposes!
>
> There's a little subtle beating in some of the 8-16ot combinations, but not so much that they don't sound like themselves. This, and the "adequate 5ths" are what I want to preserve.
>
> I want to end up with 48 pitches.
>
> Somehow, 12 pitches need to be eliminated from the current 48, but with the basic idea of 1 and 2 preserved: Slightly wide "5ths", slightly narrow "4ths", 32/27 is equivalent to 13/11.
>
> I'm going to try to figure this one out myself, but maybe someone else already has.
>
> Maybe someone would enjoy trying.
>
> (Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which has patterns of 24,8,24,8,24 mostly--except when it doesn't. There has to be a logical reason for this scale--Mr. Gene Ward Smith is too intelligent for it to be frivolous, so clearly I'm not getting it, yet.)
>
>
> ! caleb60.scl
> 60 note 13 Rube Goldberg
> 60
> !cents PC Approximate ratio
> ! 0 1/1
> 53.2 1 33/32
> 84.5 2 [4/3 below 7/5, or 21/20]
> 104.955 3 17/16
> 111.7 4 16/15
> 119.44 5 15/14
> !
> 128.3 6 14/13
> 138.6 7 13/12
> 150.6 8 12/11
> 165 9 11/10
> 179.1 10 10/9 was 182.4
> 207.2 11 9/8 wide with 3/2
> !
> 231.2 0 8/7
> 247.74 1 15/13
> 265.2 2 7/6 lowered for low 4/3
> 289.2 3 13/11 and tempered 32/27
> 313.6 4 6/5 was orginally 315.6
> 344.1 5 11/9 tempered
> !
> 359.47 6 16/13
> 385 7 5/4
> 400 8
> 414.5 9 14/11 low to go with 22/13
> 435.1 10 9/7
> 454.2 11 13/10
> !
> 470.781 0 21/16
> 496.4 1 4/3 low
> 519.551 2 27/20
> 536.95 3 15/11
> 551.3 4 11/8
> 563.4 5 18/13
> !
> 582.5 6 7/5
> 593.5 7 5/4 above tempered 9 @ 207.2
> 617.5 8 10/7
> 636.6 9 13/9
> 648.7 10 16/11
> 672 11 22/15 adjusted high to 4/3 over 10/9
> !
> 703.6 0 3/2 wide
> 729.208 1 32/21
> 745.8 2 20/13
> 764.9 3 14/9
> 772.6 4 25/16
> 782.5 5 11/7
> !
> 819 6 8/5 high, originally 813.78
> 840.53 7 13/8
> 852.6 8 18/11
> 882.7 9 5/3 lowered with 4/3
> 900 10
> 910.789 11 22/13
> !
> 933.1 0 12/7
> 952.25 1 26/15
> 968.8 2 7/4
> 992.8 3 16/9 low with 4/3
> 1013.6 4 9/5
> 1035 5 20/11
> !
> 1049.4 6 11/6
> 1061.4 7 24/13
> 1071.7 8 13/7
> 1085 9 15/8
> 1115.5 10 [4/3 above 10/7, or 40/21]
> 1146.727 11 64/33
> !
> 1200 0 2/1
>
>

Caleb, make sure that when you make scala files - all pitches are
represented in decimal form if you want to deal with cents values. That is,
if you want to represent 135 cents, make sure to write 135.0, not just 135.
That is, unless I have an older version of Scala :)

-Mike

On Mon, Aug 23, 2010 at 12:32 PM, caleb morgan <calebmrgn@...> wrote:

>
>
> Without tempering, just by removing 12 of the least-important of 60 to make
> 48:
>
> 48-pitch 13 Rube Goldberg
> 48
> !cents PC Approximate ratio
> ! 0 1/1
> 53.2 1 33/32
> 84.5 2 [4/3 below 7/5, or 21/20]
> 111.7 3 16/15
> 135 4 13/12 was 138.6
> 150.6 5 12/11
> !
> 165 6 11/10
> 179.1 7 10/9 was 182.4
> 207.2 8 9/8 wide with 3/2
> 231.2 9 8/7
> 247.74 10 15/13
> 265.2 11 7/6 lowered for low 4/3
> !
> 289.2 0 13/11 and tempered 32/27
> 313.6 1 6/5 was orginally 315.6
> 344.1 2 11/9 tempered
> 359.47 3 16/13
> 385 4 5/4
> 414.5 5 14/11 low to go with 22/13
> !
> 435.1 6 9/7
> 454.2 7 13/10
> 496.4 8 4/3 low
> 551.3 9 11/8
> 563.4 10 18/13
> 582.5 11 7/5
> !
> 617.5 0 10/7
> 636.6 1 13/9
> 648.7 2 16/11
> 672 3 22/15 adjusted high to 4/3 over 10/9
> 703.6 4 3/2 wide
> 745.8 5 20/13
> !
> 764.9 6 14/9
> 782.5 7 11/7
> 819 8 8/5 high, originally 813.78
> 840.53 9 13/8
> 852.6 10 18/11
> 882.7 11 5/3 lowered with 4/3
> !
> 910.789 0 22/13
> 933.1 1 12/7
> 952.25 2 26/15
> 968.8 3 7/4
> 992.8 4 16/9 low with 4/3
> 1013.6 5 9/5
> !
> 1035 6 20/11
> 1049.4 7 11/6
> 1061.4 8 24/13
> 1071.7 9 13/7
> 1085 10 15/8
> 1115.5 11 [4/3 above 10/7, or 40/21]
> !
> 1200 0 2/1
> On Aug 23, 2010, at 11:29 AM, calebmrgn wrote:
>
>
>
> I've been tinkering with this 60-pitch 13-limit scale, and I'm really
> pretty happy with it.
>
> But I can't help wondering whether I could get it down to 48 pitches and be
> even happier.
>
> There are several features of the scale as it currently stands that I want
> to preserve:
>
> 1) It has 8:9:10:11:12:13:14:15:16 pretty durn close on all the
> corresponding "undertones"--that is, 1/1, 16/15, 8/7, 13/8, 4/3, 16/11, 8/5,
> 16/9 (with small adjustments)
>
> 2) It has adequate chains of "5ths" and "4ths". They don't make a circle,
> and that won't bother me for my purposes, which are of course nefarious.
>
> Chain of "5ths" 1/1, 3/2, 9/8, 22/13, 14/11, 40/21, 10/7, 15/14, 8/5, 6/5,
> 9/5, 27/20 (with small adjustments)
>
> Chain of "4ths" 1/1, 4/3, 16/9, 13/11, 11/7, 21/20, 7/5, 15/8, 5/4, 5/3,
> 10/9, 22/15 (with smallish adjustments--the last 22/15 is raised
> considerably)
>
> If you look at the Scala file below, you'll see that these have often been
> adjusted to make the 4ths and 5ths adequate, if inconsistent, and neither
> chain leads back to 1/1-- which is ok for my purposes!
>
> There's a little subtle beating in some of the 8-16ot combinations, but not
> so much that they don't sound like themselves. This, and the "adequate
> 5ths" are what I want to preserve.
>
> I want to end up with 48 pitches.
>
> Somehow, 12 pitches need to be eliminated from the current 48, but with the
> basic idea of 1 and 2 preserved: Slightly wide "5ths", slightly narrow
> "4ths", 32/27 is equivalent to 13/11.
>
> I'm going to try to figure this one out myself, but maybe someone else
> already has.
>
> Maybe someone would enjoy trying.
>
> (Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which has
> patterns of 24,8,24,8,24 mostly--except when it doesn't. There has to be a
> logical reason for this scale--Mr. Gene Ward Smith is too intelligent for it
> to be frivolous, so clearly I'm not getting it, yet.)
>
>
> ! caleb60.scl
> 60 note 13 Rube Goldberg
> 60
> !cents PC Approximate ratio
> ! 0 1/1
> 53.2 1 33/32
> 84.5 2 [4/3 below 7/5, or 21/20]
> 104.955 3 17/16
> 111.7 4 16/15
> 119.44 5 15/14
> !
> 128.3 6 14/13
> 138.6 7 13/12
> 150.6 8 12/11
> 165 9 11/10
> 179.1 10 10/9 was 182.4
> 207.2 11 9/8 wide with 3/2
> !
> 231.2 0 8/7
> 247.74 1 15/13
> 265.2 2 7/6 lowered for low 4/3
> 289.2 3 13/11 and tempered 32/27
> 313.6 4 6/5 was orginally 315.6
> 344.1 5 11/9 tempered
> !
> 359.47 6 16/13
> 385 7 5/4
> 400 8
> 414.5 9 14/11 low to go with 22/13
> 435.1 10 9/7
> 454.2 11 13/10
> !
> 470.781 0 21/16
> 496.4 1 4/3 low
> 519.551 2 27/20
> 536.95 3 15/11
> 551.3 4 11/8
> 563.4 5 18/13
> !
> 582.5 6 7/5
> 593.5 7 5/4 above tempered 9 @ 207.2
> 617.5 8 10/7
> 636.6 9 13/9
> 648.7 10 16/11
> 672 11 22/15 adjusted high to 4/3 over 10/9
> !
> 703.6 0 3/2 wide
> 729.208 1 32/21
> 745.8 2 20/13
> 764.9 3 14/9
> 772.6 4 25/16
> 782.5 5 11/7
> !
> 819 6 8/5 high, originally 813.78
> 840.53 7 13/8
> 852.6 8 18/11
> 882.7 9 5/3 lowered with 4/3
> 900 10
> 910.789 11 22/13
> !
> 933.1 0 12/7
> 952.25 1 26/15
> 968.8 2 7/4
> 992.8 3 16/9 low with 4/3
> 1013.6 4 9/5
> 1035 5 20/11
> !
> 1049.4 6 11/6
> 1061.4 7 24/13
> 1071.7 8 13/7
> 1085 9 15/8
> 1115.5 10 [4/3 above 10/7, or 40/21]
> 1146.727 11 64/33
> !
> 1200 0 2/1
>
>
>
>

--- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:

> (Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which
> has patterns of 24,8,24,8,24 mostly--except when it doesn't. There has
> to be a logical reason for this scale--Mr. Gene Ward Smith is too
> intelligent for it to be frivolous, so clearly I'm not getting it, yet.)

Octacot itself is a temperament; what I gave was a 68 note MOS scale in octacot tuned by 150edo. That gives the 88 cent generator you may have heard about.

The point of mentioning it is:

(1) You wanted something regular which could be understood and learned

(2) You wanted a slightly wide fifth

(3) You wanted highly accurate 11/8 intervals

(4) You wanted to temper out 352/351, equating 13/11 and 32/27

(5) You wanted to temper out 896/891, equating 14/11 with 81/64

Octacot has all of these properties, though since it divides the fifth into eight parts it probably has a higher complexity to the fifth than what you wanted. It also tempers out other things, of course: 245/243, 385/384, 364/363, 1375/1372 etc.

Oh, weird. Scala seems to be a little different with every application. This works just fine with PianoTech. Was it crashing, or was it out of tune, or did your application not read the file?

Anyway, in the future, I'm happy to add the decimals.

-c

On Aug 23, 2010, at 2:45 PM, Mike Battaglia wrote:

> Caleb, make sure that when you make scala files - all pitches are represented in decimal form if you want to deal with cents values. That is, if you want to represent 135 cents, make sure to write 135.0, not just 135. That is, unless I have an older version of Scala :)
>
> -Mike
>
>
>
> On Mon, Aug 23, 2010 at 12:32 PM, caleb morgan <calebmrgn@...> wrote:
>
> Without tempering, just by removing 12 of the least-important of 60 to make 48:
>
>
> 48-pitch 13 Rube Goldberg
> 48
> !cents PC Approximate ratio
> ! 0 1/1
> 53.2 1 33/32
> 84.5 2 [4/3 below 7/5, or 21/20]
> 111.7 3 16/15
> 135 4 13/12 was 138.6
> 150.6 5 12/11
> !
> 165 6 11/10
> 179.1 7 10/9 was 182.4
> 207.2 8 9/8 wide with 3/2
> 231.2 9 8/7
> 247.74 10 15/13
> 265.2 11 7/6 lowered for low 4/3
> !
> 289.2 0 13/11 and tempered 32/27
> 313.6 1 6/5 was orginally 315.6
> 344.1 2 11/9 tempered
> 359.47 3 16/13
> 385 4 5/4
> 414.5 5 14/11 low to go with 22/13
> !
> 435.1 6 9/7
> 454.2 7 13/10
> 496.4 8 4/3 low
> 551.3 9 11/8
> 563.4 10 18/13
> 582.5 11 7/5
> !
> 617.5 0 10/7
> 636.6 1 13/9
> 648.7 2 16/11
> 672 3 22/15 adjusted high to 4/3 over 10/9
> 703.6 4 3/2 wide
> 745.8 5 20/13
> !
> 764.9 6 14/9
> 782.5 7 11/7
> 819 8 8/5 high, originally 813.78
> 840.53 9 13/8
> 852.6 10 18/11
> 882.7 11 5/3 lowered with 4/3
> !
> 910.789 0 22/13
> 933.1 1 12/7
> 952.25 2 26/15
> 968.8 3 7/4
> 992.8 4 16/9 low with 4/3
> 1013.6 5 9/5
> !
> 1035 6 20/11
> 1049.4 7 11/6
> 1061.4 8 24/13
> 1071.7 9 13/7
> 1085 10 15/8
> 1115.5 11 [4/3 above 10/7, or 40/21]
> !
> 1200 0 2/1
> On Aug 23, 2010, at 11:29 AM, calebmrgn wrote:
>
>>
>> I've been tinkering with this 60-pitch 13-limit scale, and I'm really pretty happy with it.
>>
>>
>> But I can't help wondering whether I could get it down to 48 pitches and be even happier.
>>
>> There are several features of the scale as it currently stands that I want to preserve:
>>
>> 1) It has 8:9:10:11:12:13:14:15:16 pretty durn close on all the corresponding "undertones"--that is, 1/1, 16/15, 8/7, 13/8, 4/3, 16/11, 8/5, 16/9 (with small adjustments)
>>
>> 2) It has adequate chains of "5ths" and "4ths". They don't make a circle, and that won't bother me for my purposes, which are of course nefarious.
>>
>> Chain of "5ths" 1/1, 3/2, 9/8, 22/13, 14/11, 40/21, 10/7, 15/14, 8/5, 6/5, 9/5, 27/20 (with small adjustments)
>>
>> Chain of "4ths" 1/1, 4/3, 16/9, 13/11, 11/7, 21/20, 7/5, 15/8, 5/4, 5/3, 10/9, 22/15 (with smallish adjustments--the last 22/15 is raised considerably)
>>
>> If you look at the Scala file below, you'll see that these have often been adjusted to make the 4ths and 5ths adequate, if inconsistent, and neither chain leads back to 1/1-- which is ok for my purposes!
>>
>> There's a little subtle beating in some of the 8-16ot combinations, but not so much that they don't sound like themselves. This, and the "adequate 5ths" are what I want to preserve.
>>
>> I want to end up with 48 pitches.
>>
>> Somehow, 12 pitches need to be eliminated from the current 48, but with the basic idea of 1 and 2 preserved: Slightly wide "5ths", slightly narrow "4ths", 32/27 is equivalent to 13/11.
>>
>> I'm going to try to figure this one out myself, but maybe someone else already has.
>>
>> Maybe someone would enjoy trying.
>>
>> (Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which has patterns of 24,8,24,8,24 mostly--except when it doesn't. There has to be a logical reason for this scale--Mr. Gene Ward Smith is too intelligent for it to be frivolous, so clearly I'm not getting it, yet.)
>>
>>
>> ! caleb60.scl
>> 60 note 13 Rube Goldberg
>> 60
>> !cents PC Approximate ratio
>> ! 0 1/1
>> 53.2 1 33/32
>> 84.5 2 [4/3 below 7/5, or 21/20]
>> 104.955 3 17/16
>> 111.7 4 16/15
>> 119.44 5 15/14
>> !
>> 128.3 6 14/13
>> 138.6 7 13/12
>> 150.6 8 12/11
>> 165 9 11/10
>> 179.1 10 10/9 was 182.4
>> 207.2 11 9/8 wide with 3/2
>> !
>> 231.2 0 8/7
>> 247.74 1 15/13
>> 265.2 2 7/6 lowered for low 4/3
>> 289.2 3 13/11 and tempered 32/27
>> 313.6 4 6/5 was orginally 315.6
>> 344.1 5 11/9 tempered
>> !
>> 359.47 6 16/13
>> 385 7 5/4
>> 400 8
>> 414.5 9 14/11 low to go with 22/13
>> 435.1 10 9/7
>> 454.2 11 13/10
>> !
>> 470.781 0 21/16
>> 496.4 1 4/3 low
>> 519.551 2 27/20
>> 536.95 3 15/11
>> 551.3 4 11/8
>> 563.4 5 18/13
>> !
>> 582.5 6 7/5
>> 593.5 7 5/4 above tempered 9 @ 207.2
>> 617.5 8 10/7
>> 636.6 9 13/9
>> 648.7 10 16/11
>> 672 11 22/15 adjusted high to 4/3 over 10/9
>> !
>> 703.6 0 3/2 wide
>> 729.208 1 32/21
>> 745.8 2 20/13
>> 764.9 3 14/9
>> 772.6 4 25/16
>> 782.5 5 11/7
>> !
>> 819 6 8/5 high, originally 813.78
>> 840.53 7 13/8
>> 852.6 8 18/11
>> 882.7 9 5/3 lowered with 4/3
>> 900 10
>> 910.789 11 22/13
>> !
>> 933.1 0 12/7
>> 952.25 1 26/15
>> 968.8 2 7/4
>> 992.8 3 16/9 low with 4/3
>> 1013.6 4 9/5
>> 1035 5 20/11
>> !
>> 1049.4 6 11/6
>> 1061.4 7 24/13
>> 1071.7 8 13/7
>> 1085 9 15/8
>> 1115.5 10 [4/3 above 10/7, or 40/21]
>> 1146.727 11 64/33
>> !
>> 1200 0 2/1
>>
>
>
>
>

I understand some of what you're saying, but don't understand all of it.

In particular, I understand the advantage of either having equal step-sizes, or having a scale with the number of pitches being a multiple of 12, or both.

I understand the usefulness of having as many intervals that are consonant with 1/1 as possible.

Also, the usefulness of having chains of 5ths and 4ths.

Also, at a much lower priority, pitches that land in the vicinity of 12ET.

The great advantage of the 58ET scale you suggested is that all approximations of 8:9:10:11:12 etc., or indeed any other pattern, will be the same relative to any key. It also is as good or better than 12ET in every respect.

This irregularity is the biggest disadvantage with "Rube Goldberg" 48. Otherwise I'm fairly happy with it.

What's mysterious about Octatet is that the step-sizes are different--i.e., 8,24,8, and it *seemed* (I might be wrong) that therefore many identical musical patterns would be different depending on what key you started from. In addition, it has 60 pitches, so it doesn't repeat on a standard keyboard. Also, it doesn't go 24,8,24,8,24 consistently--that is, the pattern is altered at some point (at 672 cents). Also, there are some note choices that perhaps you understand and can explain to me: For example, why have 584 and 592 cents? Why have 704 and 696 cents? What purpose do two sets of pitches so close to 4/3 and 3/2 serve?

It's probably obvious to you.

I'm sure with further study, I might be able to answer my own questions, but the scale was mysterious at first and second look.

To change gears slightly, supposing I restated my goals:

-36, 48, or 60 pitches.

-The same number of steps in the scale between the same musical intervals or ratios. That is, a 5/4 or reasonable approximation is always, say, 16 steps, or something. A "5th" is always 28 steps, or something.

-8:9:10:11:12:13:14 is pretty accurate, with some slow beating tolerated, on approximately the following starting points: 8/9, 8/10, 8/11, 8/12 (4/3), 8/13, 8/14, 8/15, 1/1.

-The scale repeats at the octave or 2/1, or is very, very close.

-The 3/2's, whether wide or narrow, are given more leeway than the 5/4's, which in turn are given more leeway than the 7's, the 11's and finally the 13's. That is, mainly, there is more accuracy the higher the prime.

-Pitches closer than around 10 cents apart are tempered out, or one is eliminated in favor of the other.

What are scales that come close to meeting these goals?

Btw, the only reason I didn't work longer on your 'epimophic' (?) scale was simply that I couldn't be sure of finding the beginning of it. Now that I've had more practice with keymaps, I should give it a second look. I simply couldn't get it to work for trivial technical reasons the last time around. The irregularities of JI scales makes the starting point (scale degree 0) easy to find.

Caleb

On Aug 23, 2010, at 3:37 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:
>
> > (Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which
> > has patterns of 24,8,24,8,24 mostly--except when it doesn't. There has
> > to be a logical reason for this scale--Mr. Gene Ward Smith is too
> > intelligent for it to be frivolous, so clearly I'm not getting it, yet.)
>
> Octacot itself is a temperament; what I gave was a 68 note MOS scale in octacot tuned by 150edo. That gives the 88 cent generator you may have heard about.
>
> The point of mentioning it is:
>
> (1) You wanted something regular which could be understood and learned
>
> (2) You wanted a slightly wide fifth
>
> (3) You wanted highly accurate 11/8 intervals
>
> (4) You wanted to temper out 352/351, equating 13/11 and 32/27
>
> (5) You wanted to temper out 896/891, equating 14/11 with 81/64
>
> Octacot has all of these properties, though since it divides the fifth into eight parts it probably has a higher complexity to the fifth than what you wanted. It also tempers out other things, of course: 245/243, 385/384, 364/363, 1375/1372 etc.
>
>

*epimorphic*

I left out the 'r'.

Having the same number of segments in successive stages.

-c

>
>
>
>
> On Aug 23, 2010, at 3:37 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:
>>
>> > (Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which
>> > has patterns of 24,8,24,8,24 mostly--except when it doesn't. There has
>> > to be a logical reason for this scale--Mr. Gene Ward Smith is too
>> > intelligent for it to be frivolous, so clearly I'm not getting it, yet.)
>>
>> Octacot itself is a temperament; what I gave was a 68 note MOS scale in octacot tuned by 150edo. That gives the 88 cent generator you may have heard about.
>>
>> The point of mentioning it is:
>>
>> (1) You wanted something regular which could be understood and learned
>>
>> (2) You wanted a slightly wide fifth
>>
>> (3) You wanted highly accurate 11/8 intervals
>>
>> (4) You wanted to temper out 352/351, equating 13/11 and 32/27
>>
>> (5) You wanted to temper out 896/891, equating 14/11 with 81/64
>>
>> Octacot has all of these properties, though since it divides the fifth into eight parts it probably has a higher complexity to the fifth than what you wanted. It also tempers out other things, of course: 245/243, 385/384, 364/363, 1375/1372 etc.
>>
>
>
>

This scale by Gene Ward Smith is excellent for my purposes. I'm going to try to see if there are any small modifications that will make it even better, but I suspect there aren't.

It may well be what I was looking for all along. I simply couldn't find the beginning. However, this morning, it's working fine, and the beginning is the first note on my keyboard.

I apologize to Gene for not being able to get it to work the first time around.

! caleb46.scl
46 note 13-limit epimorphic scale
46
!
49/48
36/35
21/20
16/15
13/12
11/10
10/9
9/8
8/7
7/6
13/11
6/5
11/9
16/13
5/4
14/11
9/7
21/16
4/3
27/20
11/8
7/5
45/32
10/7
16/11
40/27
3/2
32/21
14/9
11/7
8/5
13/8
18/11
5/3
22/13
12/7
7/4
16/9
9/5
11/6
13/7
15/8
21/11
35/18
55/28
2/1

Caleb

>
> ...your (Gene Ward Smith's) 'epimorphic' (?) scale ...I should give it a second look.

> Caleb
>
>
>
>
>
> On Aug 23, 2010, at 3:37 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:
>>
>> > (Meanwhile, I'm trying to figure out the mysteries of "Octocot"--which
>> > has patterns of 24,8,24,8,24 mostly--except when it doesn't. There has
>> > to be a logical reason for this scale--Mr. Gene Ward Smith is too
>> > intelligent for it to be frivolous, so clearly I'm not getting it, yet.)
>>
>> Octacot itself is a temperament; what I gave was a 68 note MOS scale in octacot tuned by 150edo. That gives the 88 cent generator you may have heard about.
>>
>> The point of mentioning it is:
>>
>> (1) You wanted something regular which could be understood and learned
>>
>> (2) You wanted a slightly wide fifth
>>
>> (3) You wanted highly accurate 11/8 intervals
>>
>> (4) You wanted to temper out 352/351, equating 13/11 and 32/27
>>
>> (5) You wanted to temper out 896/891, equating 14/11 with 81/64
>>
>> Octacot has all of these properties, though since it divides the fifth into eight parts it probably has a higher complexity to the fifth than what you wanted. It also tempers out other things, of course: 245/243, 385/384, 364/363, 1375/1372 etc.
>>
>
>
>

I now have three large scales that I want to live with for a while. Maybe some pitches need to be tweaked, maybe not. They are:

1) Epimorphic 46-pitch 13-limit scale originally by Gene Ward Smith, with tweaks by Caleb. (24/13, for one.)

2) Rube Goldberg variation. 48-pitch 13-limit scale with wide 5ths, from a suggestion by Margo Schulter. (Although she bears no responsibility for my errors.)

3) 58-Pitch ET. Good old equal temperament. 58 seems to get almost everything I want, and the rest I can live without.

How I deal with learning three big scales: a line of masking-tape above the keys, for Epimorphic. Multi-colored Post-It pieces for 'Rube'. A line of masking-tape below the keyboard for 58ET.

This is pretty much all I can handle.

However, as I said, perhaps some pitches need to be adjusted slightly in Rube and tweaked Epimorphic.

Here are the Scala files, in case anyone wants to try them, or in case anyone has ideas for subtle tweaks.

! caleb46.scl
46 note 13-lim tweaked epimorphic scale by G.W.Smith, mod by caleb
46
! a
49/48 bb
36/35 b
21/20 c
16/15 c#
13/12 d
11/10 d#
179.1 e 10/9 was 182.4
207.2 f 9/8 wide with 3/2
8/7 f#
7/6 g
13/11 g#
315.6 a
11/9 a#
16/13 b
385.0 c 5/4 was 386.3
14/11 c#
9/7 d
21/16 d#
496.4 e 4/3 low/narrow
27/20 f
11/8 f#
7/5 g
45/32 g#
10/7 a
16/11 a#
40/27 b
703.6 c 3/2 wide
32/21 c#
14/9 d
11/7 d#
813.7 e
13/8 f
18/11 f#
882.7 g 5/3 was orig 884.35
22/13 g#
12/7 a
7/4 a#
992.8 b 16/9 low/narrow with 4/3
1017.6
11/6 c#
24/13 d
1085.0 eb 15/8 lower--was 1088.3
21/11 e
35/18 f
55/28 f#
2/1 g

! 58et.scl
58-note equal temp
58
! 0
20.6896 1
41.38 2
62.07 3
82.76 4
103.44 5 "minor second"
!
124.14 6
144.83 7
165.52 8
186.2 9
206.9 10 "major second"
227.586 11
!
248.275 0
268.96 1
289.65 2 small minor third
310.34 3 large minor third
331.03 4
351.72 5
!
372.414 6
393.103 7 major third
413.793 8
434.483 9
455.172 10
475.862 11
!
496.55 0 (24) "4th"
517.241 1
537.931 2
558.62 3
579.31 4
600 5 (29) tritone

620.6896 6
641.38 7
662.07 8
682.76 9
703.44 10 "5th"
724.14 11
!
744.83 0
765.52 1
786.2 2
806.9 3 "minor sixth"
827.586 4
848.275 5
!
868.96 6
889.65 7 small major 6th
910.34 8 large major 6th
931.03 9
951.72 10
972.414 11

993.103 0 minor seventh
1013.793 1
1034.483 2
1055.172 3
1075.86 4
1096.55 5 major seventh
!
1117.241 6
1137.931 7
1158.62 8
1179.31 9
1200 10 octave
!

> 48-pitch 13 Rube Goldberg
> 48
> !cents PC Approximate ratio
> ! 0 1/1
> 84.5 1 [4/3 below 7/5, or 21/20]
> 111.7 2 16/15
> 119.43 3 15/14
> 135.0 4 13/12 was 138.6
> 150.6 5 12/11
> !
> 165.0 6 11/10
> 179.1 7 10/9 was 182.4
> 207.2 8 9/8 wide with 3/2
> 231.2 9 8/7
> 247.74 10 15/13
> 265.2 11 7/6 lowered for low 4/3
> !
> 289.2 0 13/11 and tempered 32/27
> 315.6 1 6/5
> 344.1 2 11/9 tempered
> 359.47 3 16/13
> 385.0 4 5/4
> 414.5 5 14/11 low to go with 22/13
> !
> 435.1 6 9/7
> 454.2 7 13/10
> 496.4 8 4/3 low
> 551.3 9 11/8
> 563.4 10 18/13
> 582.5 11 7/5
> !
> 617.5 0 10/7
> 636.6 1 13/9
> 648.7 2 16/11
> 672.0 3 22/15 adjusted high to 4/3 over 10/9
> 703.6 4 3/2 wide
> 745.8 5 20/13
> !
> 764.9 6 14/9
> 782.5 7 11/7
> 819.0 8 8/5 high, originally 813.78
> 840.53 9 13/8
> 852.6 10 18/11
> 882.7 11 5/3 lowered with 4/3
> !
> 910.789 0 22/13
> 933.1 1 12/7
> 952.25 2 26/15
> 968.8 3 7/4
> 992.8 4 16/9 low with 4/3
> 1017.6 5 9/5
> !
> 1035 6 20/11
> 1049.4 7 11/6
> 1061.4 8 24/13
> 1071.7 9 13/7
> 1085.0 10 15/8
> 1115.5 11 [4/3 above 10/7, or 40/21]
> !
> 1200 0 2/1
>>

On Aug 24, 2010, at 7:48 AM, caleb morgan wrote:

>
> This scale by Gene Ward Smith is excellent for my purposes. I'm going to try to see if there are any small modifications that will make it even better, but I suspect there aren't.
>
> It may well be what I was looking for all along. I simply couldn't find the beginning. However, this morning, it's working fine, and the beginning is the first note on my keyboard.
>
> I apologize to Gene for not being able to get it to work the first time around.
>
> ! caleb46.scl
> 46 note 13-limit epimorphic scale
> 46
> !
> 49/48
> 36/35
> 21/20
> 16/15
> 13/12
> 11/10
> 10/9
> 9/8
> 8/7
> 7/6
> 13/11
> 6/5
> 11/9
> 16/13
> 5/4
> 14/11
> 9/7
> 21/16
> 4/3
> 27/20
> 11/8
> 7/5
> 45/32
> 10/7
> 16/11
> 40/27
> 3/2
> 32/21
> 14/9
> 11/7
> 8/5
> 13/8
> 18/11
> 5/3
> 22/13
> 12/7
> 7/4
> 16/9
> 9/5
> 11/6
> 13/7
> 15/8
> 21/11
> 35/18
> 55/28
> 2/1
>
>
> Caleb

>
>
>
>
>
>
>
>>
>> ...your (Gene Ward Smith's) 'epimorphic' (?) scale ...I should give it a second look.
>
>> Caleb
>>
>>
>>
>>
>
>
>