a wide fifth at 703.6? (32/27=13/11?)

The moment is propitious, O ineffable Tuning-List sages. Without having
fully understood our last exchanges yet, or digested the contents of
your replies, I grovel before you.
Margo Schulter sent me an incredibly helpful email in which, among other
things, she mentioned the possibility of having a wide fifth, greater
than 700, or even 702.
At the same time, in designing my own 60-pitch 13-limit scale, I've
noticed that certain adjacent pitches are too close for comfort.
In particular, 32/27 at 294.1 cents and 13/11 at 289.2 cents.
And the inverse--27/16 at 905.9 cents and 22/13 at 910.789 cents.
Supposing I wanted to eliminate the difference between these two sets of
pitches.
That would put the fourth at 496.4 cents and the fifth at 703.6, if I
keep the 13/11 right where it is.
I think you sophisticates call this "tempering out" the difference. (Or
maybe not, because it's not splitting the difference.)
In this case, "tempering out" the difference between 32/27 and 13/11.
Or would it be better to split the difference?
I could also get rid of any need for 81/64, which I currently have,
because it would be too close to 14/11, which is 417.5. The "adjusted"
81/64 would be 414.4, so I wouldn't need it.
Rather than re-invent some wheel, I thought I'd just ask if there is
already a tuning like this--that has wide fifths that are then
equivalent to 11's and 13's.
Would this fifth be too wide?. ( I do love the sound of 1/1, 3/2, 9/8
dead-on!)
Here's my current JI 60-pitch scale, which I feel could use some
tweaking. You can see that the "chains of fifths" leave a lot to be
desired, currently.

! caleb60.scl60 note 13-limit somewhat Partchian scaleCents PC RATIO
! 0 1/1 53.2 1 33/32 84.5 2 [4/3 below 7/5, or
21/20]111.7 3 16/15 119.44 4 15/14 128.3 5 14/13
138.6 6 13/12150.6 7 12/11165 8 11/10182.4 9 10/9203.9 10
9/8231.2 11 8/7
247.74 0 15/13 266.9 1 7/6289.2 2 13/11294.1 3 32/27 315.6 4
6/5347.4 5 11/9
359.47 6 16/13 386.3 7 5/4 407.82 8 81/64417.5 9 14/11435.1
10 9/7454.2 11 13/10
470.781 0 21/16 498 1 4/3 519.551 2 27/20 536.95 3
15/11551.3 4 11/8563.4 5 18/13
582.5 6 7/5 590.223 7 45/32617.5 8 10/7636.6 9 13/9648.7 10
16/11 663 11 22/15
702 0 3/2729.208 1 32/21 745.8 2 20/13764.9 3 14/9772.6 4
25/16782.5 5 11/7
813.7 6 8/5 840.53 7 13/8852.6 8 18/11884.4 9 5/3905.9
10 27/16910.789 11 22/13
933.1 0 12/7952.25 1 26/15 968.8 2 7/4996.1 3 16/9 1017.6
4 9/51035 5 20/11
1049.4 6 11/61061.4 7 24/131071.7 8 13/71088.3 9 15/81115.5 10
[4/3 above 10/7, or 40/21]1146.727 11 64/33
1200 0 2/1

Here's a scala file that attempts to work this out.

It sounds pretty good.

I'll have to live with it a while. There's a couple of arbitrary notes stuck in, so it doesn't feel that solid. Also perhaps a few mistakes or omissions.

I call it Caleb's 60-note 13-limit Rube Goldberg

! caleb60.scl
60 note 13 Rube Goldberg
60
! 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 5/4 above tempered 16/9
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
315.6 4 6/5
344.1 5 11/9 tempered
!
359.47 6 16/13
386.3 7 5/4
400 8
417.5 9 14/11
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
663 11 22/15
!
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
!
813.7 6 8/5
840.53 7 13/8
852.6 8 18/11
884.4 9 5/3
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
1017.6 4 9/5
1035 5 20/11
!
1049.4 6 11/6
1061.4 7 24/13
1071.7 8 13/7
1088.3 9 15/8
1115.5 10 [4/3 above 10/7, or 40/21]
1146.727 11 64/33
!
1200 0 2/1

On Aug 21, 2010, at 11:00 AM, calebmrgn wrote:

>
> The moment is propitious, O ineffable Tuning-List sages. Without having fully understood our last exchanges yet, or digested the contents of your replies, I grovel before you.
>
> Margo Schulter sent me an incredibly helpful email in which, among other things, she mentioned the possibility of having a wide fifth, greater than 700, or even 702.
>
> At the same time, in designing my own 60-pitch 13-limit scale, I've noticed that certain adjacent pitches are too close for comfort.
>
> In particular, 32/27 at 294.1 cents and 13/11 at 289.2 cents.
>
> And the inverse--27/16 at 905.9 cents and 22/13 at 910.789 cents.
>
> Supposing I wanted to eliminate the difference between these two sets of pitches.
>
> That would put the fourth at 496.4 cents and the fifth at 703.6, if I keep the 13/11 right where it is.
>
> I think you sophisticates call this "tempering out" the difference. (Or maybe not, because it's not splitting the difference.)
>
> In this case, "tempering out" the difference between 32/27 and 13/11.
>
> Or would it be better to split the difference?
>
> I could also get rid of any need for 81/64, which I currently have, because it would be too close to 14/11, which is 417.5. The "adjusted" 81/64 would be 414.4, so I wouldn't need it.
>
> Rather than re-invent some wheel, I thought I'd just ask if there is already a tuning like this--that has wide fifths that are then equivalent to 11's and 13's.
>
> Would this fifth be too wide?. ( I do love the sound of 1/1, 3/2, 9/8 dead-on!)
>
> Here's my current JI 60-pitch scale, which I feel could use some tweaking. You can see that the "chains of fifths" leave a lot to be desired, currently.
>
>
> ! caleb60.scl
> 60 note 13-limit somewhat Partchian scale
> Cents PC 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
> 119.44 4 15/14
> 128.3 5 14/13
>
> 138.6 6 13/12
> 150.6 7 12/11
> 165 8 11/10
> 182.4 9 10/9
> 203.9 10 9/8
> 231.2 11 8/7
>
> 247.74 0 15/13
> 266.9 1 7/6
> 289.2 2 13/11
> 294.1 3 32/27
> 315.6 4 6/5
> 347.4 5 11/9
>
> 359.47 6 16/13
> 386.3 7 5/4
> 407.82 8 81/64
> 417.5 9 14/11
> 435.1 10 9/7
> 454.2 11 13/10
>
> 470.781 0 21/16
> 498 1 4/3
> 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
> 590.223 7 45/32
> 617.5 8 10/7
> 636.6 9 13/9
> 648.7 10 16/11
> 663 11 22/15
>
> 702 0 3/2
> 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
>
> 813.7 6 8/5
> 840.53 7 13/8
> 852.6 8 18/11
> 884.4 9 5/3
> 905.9 10 27/16
> 910.789 11 22/13
>
> 933.1 0 12/7
> 952.25 1 26/15
> 968.8 2 7/4
> 996.1 3 16/9
> 1017.6 4 9/5
> 1035 5 20/11
>
> 1049.4 6 11/6
> 1061.4 7 24/13
> 1071.7 8 13/7
> 1088.3 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:

> Rather than re-invent some wheel, I thought I'd just ask if there is
> already a tuning like this--that has wide fifths that are then
> equivalent to 11's and 13's.

If you are asking if there are temperaments which temper out both 896/891 (and so equate 14/11 and 81/64) and 352/351 (and so equate 22/13 and 27/16) and have a wide fifth, there are a lot of them. Close to your tuning there is in particular 58et, which has a fifth of 703.448. Unfortunately, this does not complete a circle of 58 fifths, but only 29. However, there are a lot of ways to construct a rank two temperament other than with an octave period and a fifth generator.

Other equal temperaments tempering out both are 41, 46, 80, 87, 121, 128, 145, 150, and 167, and these support a vast array of higher rank temperaments. If you insist on an octave period and a generator of a fifth, 80, 121, 128 or 167 will serve, the best choices being probably 80 or 167. But really, using all 58 notes of 58et might make the most sense for you.

Thanks, I'll try 58-et!

And, for further study, I've saved your answer to my "microtonal wisdom" file.

caleb

On Aug 21, 2010, at 2:06 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:
>
> > Rather than re-invent some wheel, I thought I'd just ask if there is
> > already a tuning like this--that has wide fifths that are then
> > equivalent to 11's and 13's.
>
> If you are asking if there are temperaments which temper out both 896/891 (and so equate 14/11 and 81/64) and 352/351 (and so equate 22/13 and 27/16) and have a wide fifth, there are a lot of them. Close to your tuning there is in particular 58et, which has a fifth of 703.448. Unfortunately, this does not complete a circle of 58 fifths, but only 29. However, there are a lot of ways to construct a rank two temperament other than with an octave period and a fifth generator.
>
> Other equal temperaments tempering out both are 41, 46, 80, 87, 121, 128, 145, 150, and 167, and these support a vast array of higher rank temperaments. If you insist on an octave period and a generator of a fifth, 80, 121, 128 or 167 will serve, the best choices being probably 80 or 167. But really, using all 58 notes of 58et might make the most sense for you.
>
>

Here's 58et in a Scala file. Should be right.

First impression: I really like this, and it's "easy to think".

That is, the regularity makes it easy to grok.

Second impression: I miss certain ratios, like 11/8, immediately.

Surprisingly, 558.82 doesn't sound at all like 551.3 to me.

Now I have two big scales to practice.

(Caleb goes away, a gaping hole where the top of his head used to be, after the explosion.)

! 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 7
951.72 9
972.414 10
993.103 11 minor seventh
!
1013.793 0
1034.483 1
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

On Aug 21, 2010, at 2:23 PM, caleb morgan wrote:

> Thanks, I'll try 58-et!
>
>
> And, for further study, I've saved your answer to my "microtonal wisdom" file.
>
> caleb
>
>
> On Aug 21, 2010, at 2:06 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:
>>
>> > Rather than re-invent some wheel, I thought I'd just ask if there is
>> > already a tuning like this--that has wide fifths that are then
>> > equivalent to 11's and 13's.
>>
>> If you are asking if there are temperaments which temper out both 896/891 (and so equate 14/11 and 81/64) and 352/351 (and so equate 22/13 and 27/16) and have a wide fifth, there are a lot of them. Close to your tuning there is in particular 58et, which has a fifth of 703.448. Unfortunately, this does not complete a circle of 58 fifths, but only 29. However, there are a lot of ways to construct a rank two temperament other than with an octave period and a fifth generator.
>>
>> Other equal temperaments tempering out both are 41, 46, 80, 87, 121, 128, 145, 150, and 167, and these support a vast array of higher rank temperaments. If you insist on an octave period and a generator of a fifth, 80, 121, 128 or 167 will serve, the best choices being probably 80 or 167. But really, using all 58 notes of 58et might make the most sense for you.
>>
>
>
>

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> Here's 58et in a Scala file. Should be right.
>
> First impression: I really like this, and it's "easy to think".
>
> That is, the regularity makes it easy to grok.
>
> Second impression: I miss certain ratios, like 11/8, immediately.
>
> Surprisingly, 558.82 doesn't sound at all like 551.3 to me.

As I said, there are other possibilities. For example, there is octacot temperament. 41 or 68 notes are possibilities, and the 11/8 is excellent.

! octa68.scl
Octacot[68] in 150edo
68
!
24.0
32.0
56.0
64.0
88.0
112.0
120.0
144.0
152.0
176.0
200.0
208.0
232.0
240.0
264.0
288.0
296.0
320.0
328.0
352.0
376.0
384.0
408.0
416.0
440.0
464.0
472.0
496.0
504.0
528.0
552.0
560.0
584.0
592.0
616.0
640.0
648.0
672.0
696.0
704.0
728.0
736.0
760.0
784.0
792.0
816.0
824.0
848.0
872.0
880.0
904.0
912.0
936.0
960.0
968.0
992.0
1000.0
1024.0
1048.0
1056.0
1080.0
1088.0
1112.0
1136.0
1144.0
1168.0
1176.0
1200.0