Full thread (6 messages)
From: calebmrgn (2010-08-21)
Subject: 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
From: caleb morgan (2010-08-21)
Subject: Re: [tuning] a wide fifth at 703.6? (32/27=13/11?)
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
>
>
>
From: genewardsmith (2010-08-21)
Subject: Re: a wide fifth at 703.6? (32/27=13/11?)
--- 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.
From: caleb morgan (2010-08-21)
Subject: Re: [tuning] Re: a wide fifth at 703.6? (32/27=13/11?)
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.
>
>
From: caleb morgan (2010-08-21)
Subject: Re: [tuning] Re: a wide fifth at 703.6? (32/27=13/11?)
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.
>>
>
>
>
From: genewardsmith (2010-08-22)
Subject: Re: a wide fifth at 703.6? (32/27=13/11?)
--- 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