Topic: a wide fifth at 703.6? (32/27=13/11?)
1 scales
| File | Description | Notes | Period (ยข) |
|---|---|---|---|
| octa68 | Octacot[68] in 150edo | 68 | 1200.0 |
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