Yahoo Tuning Groups Ultimate Backup tuning Compromise between 46EDO & Gene Ward Smith's 46 Epimorphic?

Help, o wise ones.

46EDO is nearly perfect, Gene Ward Smith's epimorphic Just scale is nearly perfect.

Can I somehow split the difference to make something better than either?

I've tweaked GWS's epi scale to make it even more to my liking.

I include 46EDO and my tweaked GWS scale below.

All the important 8:9:10:11:12:13:14:15:16 approximations sound good,

and on a standard keyboard, they go 8,7,6,6,5,5,5,4 steps apart.

These are the two essential things to keep that I like.

48 pitches per 2/1 is my maximum practical limit, so 46 is fine.

The only problem with the current tweaked scale, below, is that other intervallic patterns
have slightly inconsistent patterns of keys. Such as, a chain of 4ths from 15/14, or C#, if the scale begins on A. The 4/3 interval *should* be 19 tones apart, or an octave plus 7 keys. It isn't.

Is it possible to make a subtle tweak to my mod2 version so that the fingering is slightly more consistent?

I want to keep the basic pitch relations I've got, but make just make the key patterns slightly more consistent.

Not every starting point needs to have "good" 4ths or 5ths. Just the important ones.

The tweak might be impossible. Or it might involve adjusting a few pitches, or adding or subtracting a few.

I'd even accept a tweak with a repetition of the identical pitch on two adjacent keys, if that would make the key-patterns more consistent.

What's *not" important is consistent steps between adjacent keys, that is, the small intervals between adjacent keys don't need to be exactly the same.

What *is* important: chains of 3/2 approximations, 4/3 approx., and 8:9:10:11:12:13:14:15:16s, with leeway allowed in the 11s, 13s and 15s.

The desired result is close to 46 EDO or some other EDO perhaps, but is a sort of large "well" temperament, perhaps. It has better 7's and 11's and 13's than 46 EDO--being closer to JI.

Is this simply impossible by definition?

Here are my two starting-points, below.

! 46et.scl
46-note equal temp
46
! 0
26.08696 1
52.17391 2
78.261 3
104.34 4 b2
130.435 5
156.52 6
182.609 7
208.696 8 2
234.783 9
260.8696 10
286.9565 11 b3 narrow
!
313.0435 12 b3 wide
339.13 13
365.2174 14
391.3043 15 3
417.3913 16
443.4783 17
469.5652 18
495.6522 19 4 !4th= 1 oct + fifth
521.7391 20
547.8261 21
573.913 22
600 23 #4/b5 middle-!tritone= 1 oct + major seventh-----------------------------------
!
626.087 24
652.1739 25
678.2609 26
704.3478 27 5 !fifth= 2 oct + minor third
730.4348 28
756.5217 29
782.6087 30
808.6957 31 b6
834.7826 32
860.8696 33
886.9565 34
913.0435 35 6
!
939.1304 36
965.2174 37
991.3043 38
1017.391 39
1043.478 40
1069.565 41
1095.652 42
1121.739 43
1147.826 44
1173.913 45
1200 46

! caleb46.scl
46 note 13-lim tweaked epiby G.W.Smith, mod2 by caleb
46
! a ************keep 8,7,6,6,5,5,5,4,
30
53. 4th below 11/8, 5th above 11/8
21/20 c 84.46 ********keep
15/14 c# 119.4 (was 16/15 in previous)*************
13/12 d 138.57
150.68 12/11 d# changed to give "5th" above /11
179.1 e 10/9 was 182.4 *******keep
207.2 f 9/8 wide with 3/2 *********keep
8/7 f# 231
7/6 g 266
13/11 g# 289 32/27 is 294.13 *************keep
!
313.6 a was 6/5 315.6 ************keep
11/9 a# 347.4
16/13 b
385.0 c 5/4 was 386.3 ************
14/11 c# 417.5
9/7 d 435
21/16 d# 470.8
496.4 e 4/3 low/narrow
27/20 f 519.55
11/8 f# 551.3
7/5 g 582 *********keep
593
!
!
10/7 a 617.5 ********keep
648.68 16/11 a#
40/27 b 4th above our 10/9 is 677, 22/15 is 663, 40/27 is 680.4
703.6 3/2 wide *******keep
32/21 c#
14/9 d 764
11/7 d# 782.49 keep**************
819. e high 8/5, keep8***************
13/8 f 840.52
18/11 f# 852
882.7 g 5/3 was orig 884.35 ********keep
22/13 g# 911
!
12/7 a 933
7/4 a# 968
992.8 b 16/9 low/narrow with 4/3, was 996 ************
1013.6 ********keep, end of +5ths chain, was 9/5, 1017.59
11/6 c#
24/13 d
1085.0 eb 15/8 lower--was 1088.3 ***********keep
40/21 1115.5 *******keep
1147
1170
2/1 g

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

Sorry 'bout that! The file below should work better.

I changed 45/32 to be higher @ 605 cents, from 590.22.

! keep 8,7,6,6,5,5,5,4, on "important" tonalities, ! keep 19 steps to a 4th, keep "wide, high" 5ths, allow some inconsistencies where less important--somehow! (or is that impossible?)

I guess the way I understand it is, is there any wiggle room to get closer to JI, while still having good chains of 4ths and 5ths, and consistent fingering?

And, if I were patient enough, (which I am) how to search for other things that are very close to this?

You, Gene, were the one who gave me the idea of 46-note JI, and also you've composed in 46EDO.

Is there some tuning that sort of splits the difference in a way that's better than either?

Caleb

! caleb46.scl
46 note tweaked epimorphic scale by G.W.Smith, mod by caleb
46
! a
35.69
48.77
84.46
119.4
138.572
150.68
179.1
207.2
231.17
266.87
289.2
!
313.6
347.4
359.47
385.0
417.5
435.1
470.78
496.4
519.55
551.31
582.5
605.0
!
!
617.5
648.68
680.45
703.6
729.22
764.9
782.49
819.0
840.53
852.6
882.7
910.79
!
933.12
968.82
992.8
1013.6
1049.36
1061.43
1085.0
1115.53
1151.23
1168.8
1200.0

On Sep 4, 2010, at 4:04 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:
> >
> >
> > ! caleb46.scl
> > 46 note 13-lim tweaked epiby G.W.Smith, mod2 by caleb
>
> You've got all kinds of comments which prevent it from being a valid scl file. Could you post a version with only either a rational number or a floating point number (representing cents) on each line? You can also place comments after a ! on a line.
>
>

🔗caleb morgan <calebmrgn@...>

For example, what would happen if you divided 7/5 in 23 equal parts, then divided 10/7 in 23 equal parts?

Probably nothing good, but I'm just trying to brainstorm.

Caleb

On Sep 4, 2010, at 4:40 PM, caleb morgan wrote:

> Sorry 'bout that! The file below should work better.
>
>
> I changed 45/32 to be higher @ 605 cents, from 590.22.
>
> ! keep 8,7,6,6,5,5,5,4, on "important" tonalities, ! keep 19 steps to a 4th, keep "wide, high" 5ths, allow some inconsistencies where less important--somehow! (or is that impossible?)
>
> I guess the way I understand it is, is there any wiggle room to get closer to JI, while still having good chains of 4ths and 5ths, and consistent fingering?
>
> And, if I were patient enough, (which I am) how to search for other things that are very close to this?
>
> You, Gene, were the one who gave me the idea of 46-note JI, and also you've composed in 46EDO.
>
> Is there some tuning that sort of splits the difference in a way that's better than either?
>
> Caleb
>
>
>
>
>
> ! caleb46.scl
> 46 note tweaked epimorphic scale by G.W.Smith, mod by caleb
> 46
> ! a
> 35.69
> 48.77
> 84.46
> 119.4
> 138.572
> 150.68
> 179.1
> 207.2
> 231.17
> 266.87
> 289.2
> !
> 313.6
> 347.4
> 359.47
> 385.0
> 417.5
> 435.1
> 470.78
> 496.4
> 519.55
> 551.31
> 582.5
> 605.0
> !
> !
> 617.5
> 648.68
> 680.45
> 703.6
> 729.22
> 764.9
> 782.49
> 819.0
> 840.53
> 852.6
> 882.7
> 910.79
> !
> 933.12
> 968.82
> 992.8
> 1013.6
> 1049.36
> 1061.43
> 1085.0
> 1115.53
> 1151.23
> 1168.8
> 1200.0
>
>
>
> On Sep 4, 2010, at 4:04 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, "calebmrgn" <calebmrgn@...> wrote:
>> >
>> >
>> > ! caleb46.scl
>> > 46 note 13-lim tweaked epiby G.W.Smith, mod2 by caleb
>>
>> You've got all kinds of comments which prevent it from being a valid scl file. Could you post a version with only either a rational number or a floating point number (representing cents) on each line? You can also place comments after a ! on a line.
>>
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

Thanks!, I'm checking them out now.

Will have something to say tomorrow.

-c

On Sep 4, 2010, at 6:00 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > I guess the way I understand it is, is there any wiggle room to get closer to JI, while still having good chains of 4ths and 5ths, and consistent fingering?
>
> It's not what you seemed to be asking for, but below are two scales I derived from yours. The wide fifths are now narrow ones, and 13 is a little screwed over, but there are advantages also. Maybe you can use it as another starting point, or see if it gives you ideas.
>
> ! caleb46_4.scl
> caleb46 re-tweaked
> 46
> !
> 34.1041
> 51.1932
> 83.0422
> 117.8827
> 133.9278
> 150.5304
> 184.4619
> 201.5270
> 234.4075
> 267.4520
> 285.5358
> 315.1545
> 350.9402
> 365.5360
> 384.7127
> 417.5425
> 433.6369
> 467.8656
> 501.1001
> 517.6815
> 551.1279
> 582.6132
> 602.9948
> 618.2099
> 651.4617
> 682.860
> 700.5025
> 733.4973
> 768.0480
> 784.1441
> 818.4160
> 835.7333
> 850.7061
> 885.6155
> 917.1322
> 934.7707
> 967.3138
> 1000.3701
> 1016.1738
> 1050.3968
> 1067.8418
> 1085.1891
> 1117.9017
> 1152.6154
> 1167.7300
> 1200.0000
>
> ! caleb44.scl
> caleb46 massacred
> 44
> !
> 33.3285
> 49.1413
> 82.5632
> 116.9509
> 133.7225
> 149.7450
> 183.7235
> 199.9940
> 232.9493
> 266.1797
> 284.0458
> 314.9424
> 350.1557
> 383.6852
> 416.3771
> 433.0033
> 466.3924
> 499.8415
> 516.4081
> 550.5131
> 582.7073
> 600.3996
> 617.0051
> 649.5010
> 683.3140
> 699.7916
> 733.0869
> 766.9046
> 783.3466
> 816.2088
> 849.5745
> 884.3029
> 915.4063
> 933.4554
> 966.7936
> 999.6010
> 1016.0054
> 1050.0168
> 1066.1453
> 1082.7801
> 1117.1462
> 1150.7489
> 1166.3940
> 1200.0000
>
>

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

On 5 September 2010 06:24, genewardsmith <genewardsmith@...> wrote:

> The most obvious way to do this is to use a 46 note MOS of an appropriate
> rank two temperament such as the 17-limit versions of wizard, valentine,
> rodan or diaschismic. Wizard would be excellent but for the fact that you want
> long chains of wide fifths, so it seems to me rodan or diaschismic would be
> the best choices. I could give 46 note MOS for those if you want them.

Why 17? I don't see any mention of 17 in the thread above. Anyway,
I'll give URLs to the automatically generated pages, in the hope that
they'll be understood. This is diaschismic, which is an obvious
choice if you want the fifths:

http://tinyurl.com/29zs84h

Rodan:

http://tinyurl.com/258ngvy

Valentino, which must be the Valentine variant:

http://tinyurl.com/23h89ye

Wizard only comes up in the 17-limit search, so here's that:

http://tinyurl.com/2bnlamt

Unidec looks better, but may or may not be so in reality.

13-limit:

http://tinyurl.com/23cno9d

17-limit:

http://tinyurl.com/26oodsl

You can take any rank 2 temperament and smooth the edges to get a well
temperament.

Graham

🔗caleb morgan <calebmrgn@...>

Tuning lightweight Caleb is overwhelmed by talking to the heavy cats.

Thank you both, and, really, I'm embarrassed to admit I'm not sure if I understand, but I'm determined to understand this.

I'd be delighted with chains of 5ths either slightly wide or slightly narrow, and with something approximating a 13-limit. 17 isn't necessary. I like the octave slightly off--not too much, though.

I do seem to prefer the sound of higher fifths. I've also found that I prefer a *little* detuning or beating to absolutely perfect JI, but I prefer something less consistent and closer to JI than 46 EDO.

Because of checking out some of the scales by GWS and others, I've come to appreciate the importance of regular fingering patterns. Regular fingering patterns are more important than repeating at some multiple of 12, which I *formerly* thought was one way to make a scale easier to learn. Easy learning becomes increasingly important as more pitches are added.

I will learn about the theory by generating these tunings, I hope.

I can't plug them into Scala, because I don't have that running on my Mac, but I hope to in the future. Therefore, I can't follow the instructions literally, as given on one of your pages.

However, let me guess, and try using LMSO, and a calculator.

Let's take them one at a time.

http://tinyurl.com/29zs84h

Diaschismic

I use 103.608 as a generator inside a modulus of 599.447 for 23 iterations, then take each of those cent values and add 599.447 cents to get all 46? Or do I add *600* cents?

Rodan

I use 234.48 inside a modulus of 1199.989 up to 46?

Valentino

I use 77.971 inside a mod of 1200.201 etc.?

Unidec

I do 23 iterations of 183.315 inside 600.383, then add 600.383 for the remaining 23? Or do I add 600?

Unidec one dimension higher

23 interations of 183.318 inside 600.4, then add 600.4 for the remaining 23? Or do I add 600?

I'm sorry you have to be so literal and concrete, but that's my level.

If I may hazard a very small joke: when it comes to your mastery of tuning, *both* of you are two dimensions higher.

I'm exhausted right now, I'll work on this tomorrow.

Thanks, Caleb

On Sep 4, 2010, at 7:29 PM, Graham Breed wrote:

> On 5 September 2010 06:24, genewardsmith <genewardsmith@...> wrote:
>
> > The most obvious way to do this is to use a 46 note MOS of an appropriate
> > rank two temperament such as the 17-limit versions of wizard, valentine,
> > rodan or diaschismic. Wizard would be excellent but for the fact that you want
> > long chains of wide fifths, so it seems to me rodan or diaschismic would be
> > the best choices. I could give 46 note MOS for those if you want them.
>
> Why 17? I don't see any mention of 17 in the thread above. Anyway,
> I'll give URLs to the automatically generated pages, in the hope that
> they'll be understood. This is diaschismic, which is an obvious
> choice if you want the fifths:
>
> http://tinyurl.com/29zs84h
>
> Rodan:
>
> http://tinyurl.com/258ngvy
>
> Valentino, which must be the Valentine variant:
>
> http://tinyurl.com/23h89ye
>
> Wizard only comes up in the 17-limit search, so here's that:
>
> http://tinyurl.com/2bnlamt
>
> Unidec looks better, but may or may not be so in reality.
>
> 13-limit:
>
> http://tinyurl.com/23cno9d
>
> 17-limit:
>
> http://tinyurl.com/26oodsl
>
> You can take any rank 2 temperament and smooth the edges to get a well
> temperament.
>
> Graham
>

🔗Graham Breed <gbreed@...>

On 5 September 2010 08:35, caleb morgan <calebmrgn@...> wrote:

> Because of checking out some of the scales by GWS and others, I've come to appreciate the importance of regular fingering patterns. Regular fingering patterns are more important than repeating at some multiple of 12, which I *formerly* thought was one way to make a scale easier to learn. Easy learning becomes increasingly important as more pitches are added.

I still think 46 is a lot of notes if you don't have a generalized
keyboard. But, given that, regular temperaments will have this
consistency.

> I will learn about the theory by generating these tunings, I hope.
> I can't plug them into Scala, because I don't have that running on my Mac, but I hope to in the future. Therefore, I can't follow the instructions literally, as given on one of your pages.
> However, let me guess, and try using LMSO, and a calculator.
> Let's take them one at a time.
> http://tinyurl.com/29zs84h
> Diaschismic
> I use 103.608 as a generator inside a modulus of 599.447 for 23 iterations, then take each of those cent values and add 599.447 cents to get all 46? Or do I add *600* cents?

Yes. I don't know LMSO, hopefully it makes this easy. You should add
599.447 cents.

> Rodan
> I use 234.48 inside a modulus of 1199.989 up to 46?
> Valentino
> I use 77.971 inside a mod of 1200.201 etc.?

Yes.

> Unidec
> I do 23 iterations of 183.315 inside 600.383, then add 600.383 for the remaining 23? Or do I add 600?

You add 600.383 but it won't make a great amount of difference if you add 600.

> Unidec one dimension higher
> 23 interations of 183.318 inside 600.4, then add 600.4 for the remaining 23? Or do I add 600?

It's the same as the other Unidec, really, but with a mapping for
intervals of 17.

> I'm sorry you have to be so literal and concrete, but that's my level.

That's exactly what you have to do, and what the readouts are telling
you to do, so I don't see anything to apologize for. You can also use
the mappings to get the fingering patterns. The one for 46 should
always be the same. There's another by generator that tells you which
intervals are in tune. You can also use this to work out how many
pure fifths you get.

Graham

🔗Michael <djtrancendance@...>

🔗caleb morgan <calebmrgn@...>

Amazing!
I'm starting to get how this works.

"so it seems to me rodan or diaschismic would be
> the best choices. I could give 46 note MOS for those if you want them."

I know that MOS stands for "moment of symmetry". But I don't know what these would be in practice, yet. So, yes, I'd be very curious in the MOS scales.

Would a 46-note MOS be a subset of a larger EDO?

Would these be substantially different from Rodan, Valentino, and Diaschismic?

I didn't like Unidec, I think, because the 9/8 approximation was so high--I could be wrong.

Same thing with Wizard, iirc, the 9/8 was too high.

The other scale I really liked, but for slightly weird layout and/or 5ths, was GWS's "Cal Massacre"--which I renamed "Masaka", to make it sound, um, soulful.

Ok, can we have something a little closer to JI in spirit, 11-limit, more unequal step-sizes (that is, both in terms of number of different sizes and relative proportion of sizes)
a chain of 5ths at least 10 notes long, and 4ths 10 notes long, regular layout for 4ths and 5ths, and 36 to 46 pitches, with 9/8's pretty close?

The result would only be really in tune or consistent in 8 to 12 keys, perhaps.

It sounds sort of childishly demanding to put it that way. But you guys already know how to do this stuff without breaking a sweat.

For those who are interested, I give the Scala files of the ones I liked.

I'm still a long way from being able to tell these apart by ear.

Caleb

(Thanks--you guys really are amazing.)

! diachismic 46 (GB)
46-n diachismic
46
!0.,
22.202,
59.205,
81.406,
103.608,
125.81,
162.813,
185.014,
207.216,
229.418,
266.421,
288.622,
!
310.824,
333.026,
370.029,
392.23,
414.432,
436.634,
473.637,
495.839,
518.04,
540.242,
577.245
599.447
!
!
621.649
658.652
680.853
703.055
725.3
762.4
784.5
806.7
828.8
865.9
888.1
910.3
!
932.5
969.7
991.8
1013.9
1036.1
1073.2
1095.3
1117.1
1139.7
1176.8
1198.89

! cal46.scl
46-n 4 Cal by G.W.Smith "required"
46
!
22.9251
56.2576
80.6369
104.0612
126.6171
160.4026
184.7357
208.1710
231.9308
263.2192
288.7711
!
311.2289
336.7808
368.0692
391.8290
415.2643
439.5974
473.3829
495.9388
519.3631
543.7424
577.0749
600.0000
!
!
622.9251
656.2576
680.6369
704.0612
726.6171
760.4026
784.7357
808.1710
831.9308
863.2192
888.7711
911.2289
!
936.7808
968.0692
991.8290
1015.2643
1039.5974
1073.3829
1095.9388
1119.3631
1143.7424
1177.0749
1200.0000

! Rodan 46 (GB)
46-n 234.48 generator ca. 28&14cent steps
46
!0.,
27.586,
55.172,
82.758,
110.344,
124.137,
151.723,
179.309,
206.895,
234.481,
262.066,
289.652,
!
317.238,
344.824,
358.617,
386.203,
413.789,
441.375,
468.961,
496.547,
524.133,
551.719,
579.305,
593.098,
!
!
620.684,
648.27,
675.856,
703.442,
731.028,
758.613,
786.199,
813.785,
841.371,
855.164,
882.75,
910.336,
!
937.922,
965.508,
993.094,
1020.68,
1048.266,
1075.852,
1089.645,
1117.231,
1144.817,
1172.403
1199.989

! Valentino 46 (GB)
46-n, 77.971 gen, 30.6 & 16.7-cents
46
!0.,
30.636,
47.335,
77.971,
108.607,
125.306,
155.942,
186.578,
203.277,
233.913,
264.549,
281.248,
!
311.884,
342.52,
359.219,
389.855,
420.491,
437.19,
467.826,
498.462,
515.161,
545.797,
576.433,
593.132,
!
!
623.768,
654.404,
685.04,
701.739,
732.375,
763.011,
779.71,
810.346,
840.982,
857.681,
888.317,
918.953,
!
935.652,
966.288,
996.924,
1013.623,
1044.259,
1074.895,
1091.594,
1122.23,
1152.866,
1169.565
1200.201

On Sep 4, 2010, at 7:29 PM, Graham Breed wrote:

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

I apologize for being more greedy than competent.

It's exciting to have tools for designing and hearing these scales, so I'm trying things so fast that many of my ideas are barely coherent.

I also apologize if I sound like I'm treating you like Genie Ward Smith, Granter of Tuning Wishes.

It's just that I don't yet have a clear sense of what I want to hear, and how to capture that in as few a set of notes as possible, with a consistent pattern of keys.

Let's say we brought the limit down to 11.

I certainly like to hear the difference between 11:10, 10:9, 9:8, 8:7, 7:6, and 5:4.

It seems that to satisfy what I want to hear, the scale has to have something close to 182¢, 204¢,
231¢, 266¢, something to capture the sound of 6:5 and 5:4, something to get 11/8.

So, what is close to the fewest notes that gives complete 7:8:9:10:11:12 on 8/7, 1/1, 16/9, 8/5, 16/11, 4/3, 8/7, 16/15 in EDO or JI?

(Maybe I don't need a 13-limit, but I do really like the sound of that interval--13/2, 13/4, 13/8.)

Supposing we left it out and used just 5/3 and 27/16, and also included a 15/8--that is, some OTs of 3.

What EDOs come close?

Also, I'm curious how to generate rank 3 and rank 4 temperaments when I stumble on them.

For example, what to do with this:

Portent one dimension higher

Equal Temperament Mappings
2 3 5 7 11 13
[< 31 49 72 87 107 115 ]
< 41 65 95 115 142 152 ]
< 46 73 107 129 159 170 ]>

Reduced Mapping
2 3 5 7 11 13
[< 1 1 0 3 5 7 ]
< 0 3 0 -1 4 -5 ]
< 0 0 1 0 -1 -1 ]>

Generator Tunings (cents)
[1200.067, 234.186, 2786.754>

Step Tunings (cents)
[5.660, 11.068, 12.409>

Tuning Map (cents)
<1200.067, 1902.624, 2786.754, 3366.014, 4150.323, 4442.783]

a generator of 234.186 repeated 41 times makes step sizes of 29.07 and 30.696 if the modulus is 1200 cents.

(This looks pretty good, actually: 0., 29.07, 58.14, 87.21, 116.28, 146.976, 176.046, 205.116, 234.186, 263.256, 292.326, 321.396, 350.466, 381.162, 410.232, 439.302, 468.372, 497.442, 526.512, 555.582, 584.652, 615.348, 644.418, 673.488, 702.558, 731.628, 760.698, 789.768, 818.838, 849.534, 878.604, 907.674, 936.744, 965.814, 994.884, 1023.954, 1053.024, 1083.72, 1112.79, 1141.86, 1170.93

But I don't understand what to do with the 2786 info (5th partial) that's included under Generator Tunings.

This is ?Rank 2 because of 29 cent and 30.6 cent step-sizes?

Caleb

On Sep 5, 2010, at 2:44 PM, genewardsmith wrote:

>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > I know that MOS stands for "moment of symmetry". But I don't know
> what these would be in practice, yet. So, yes, I'd be very curious in
> the MOS scales.
>
> You seem to have them already.
>
> > Would a 46-note MOS be a subset of a larger EDO?
>
> If you used that tuning. Do you want it to be?
>
>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

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

> I certainly like to hear the difference between 11:10, 10:9, 9:8, 8:7, 7:6, and 5:4.

See what you think of this:

! newts.scl
!
11-limit scale with boatload of neutral thirds
41
!
32.91065
50.80017
83.50287
116.45974
149.89384
182.87269
199.63490
233.54070
266.22566
299.83415
316.84989
349.84324
383.18550
400.34913
449.39797
466.56160
499.90386
532.89721
549.91295
583.52144
616.20640
650.11220
666.87441
699.85326
733.28736
766.24423
798.94693
816.83645
849.74710
883.17168
899.88473
933.63277
966.38375
999.89157
1024.87355
1049.85553
1083.36335
1116.11433
1149.86237
1166.57542
1200.00000

> What EDOs come close?

46 or 58 don't do it?

> Also, I'm curious how to generate rank 3 and rank 4 temperaments when I stumble on them.
>
> For example, what to do with this:
>
> Portent one dimension higher

I have portent written up here:

http://xenharmonic.wikispaces.com/Gamelismic+family

but it probably won't make much sense. I'm trying to think of what a good introductory page on these writeups of rank 3 temperaments should look like. But one thing you might glean from it is that constructing scales by taking products of 8/7 and 12/11 and octave reducing, then tempering via a portent tuning map, would be a way of getting portent scales.

🔗caleb morgan <calebmrgn@...>

Thank you!

I'll listen today.

Nah, I really like 41, 46, and 58EDO.

I'm experimenting with different patterns (srutis) within those, to see if I can reduce the number of pitches and end up with any kind of keyboard mapping that is consistent, or at least propitious on a standard keyboard.

You may already know that this is possible or impossible, but it's teaching me a lot.

The holy grail for today would be some sruti pattern of 41, 46 or 58 EDO that contains the pitches I like, plus some kind of consistency in fingering. It might be possible, might not.

I realized that any member (value in cents) of 41 can be a generator, because 41 is a prime number.

Probably any *other* member of 46 or 58 can be a generator, because 23 and 39 are prime.

I was sort of hoping that by finding the right generator within one of these three EDOs, that there would be a particularly effective pattern of srutis--or, pitches included or omitted.

Failing that, I'm simply going to take inconsistent subsets of those EDOs with arbitrary patterns, and see what those are like to play with.

As a result of these experiments and this list, I now have upwards of 30 different tunings/scales,
ranging in # of pitches between 27 and 58, with 4 or 5 JI versions, 5 or 6 scales by Mr. Gene Ward Smith, the Wendy Carlos Gamma scale, and many variations on 46--Diaschismic, Rodan, Valentino.

At some point soon, I'm going to switch to investigating dynamic retuning with Little Miss Scale Oven, and also, composing.

Mustn't forget the composing part.

Thanks again,

Caleb

On Sep 6, 2010, at 4:41 AM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > I certainly like to hear the difference between 11:10, 10:9, 9:8, 8:7, 7:6, and 5:4.
>
> See what you think of this:
>
> ! newts.scl
> !
> 11-limit scale with boatload of neutral thirds
> 41
> !
> 32.91065
> 50.80017
> 83.50287
> 116.45974
> 149.89384
> 182.87269
> 199.63490
> 233.54070
> 266.22566
> 299.83415
> 316.84989
> 349.84324
> 383.18550
> 400.34913
> 449.39797
> 466.56160
> 499.90386
> 532.89721
> 549.91295
> 583.52144
> 616.20640
> 650.11220
> 666.87441
> 699.85326
> 733.28736
> 766.24423
> 798.94693
> 816.83645
> 849.74710
> 883.17168
> 899.88473
> 933.63277
> 966.38375
> 999.89157
> 1024.87355
> 1049.85553
> 1083.36335
> 1116.11433
> 1149.86237
> 1166.57542
> 1200.00000
>
> > What EDOs come close?
>
> 46 or 58 don't do it?
>
> > Also, I'm curious how to generate rank 3 and rank 4 temperaments when I stumble on them.
> >
> > For example, what to do with this:
> >
> > Portent one dimension higher
>
> I have portent written up here:
>
> http://xenharmonic.wikispaces.com/Gamelismic+family
>
> but it probably won't make much sense. I'm trying to think of what a good introductory page on these writeups of rank 3 temperaments should look like. But one thing you might glean from it is that constructing scales by taking products of 8/7 and 12/11 and octave reducing, then tempering via a portent tuning map, would be a way of getting portent scales.
>
>

🔗caleb morgan <calebmrgn@...>

🔗Michael <djtrancendance@...>

🔗Graham Breed <gbreed@...>

On 6 September 2010 05:55, caleb morgan <calebmrgn@...> wrote:

> Let's say we brought the limit down to 11.
> I certainly like to hear the difference between 11:10, 10:9, 9:8, 8:7, 7:6, and 5:4.

Do you care about 12:11? If you want that distinct as well, what
you'd be looking for is 11-limit uniqueness. I think Miracle does it
first. It's not something the new code tells you because I left odd
limits behind. But I only unlinked the old code, I didn't delete it.
So this tells you Miracle is unique:

http://x31eq.com/cgi-bin/temperament.cgi?et1=31&et2=41&limit=11

This tells you Orwell isn't:

http://x31eq.com/cgi-bin/temperament.cgi?et1=31&et2=22&limit=11

> It seems that to satisfy what I want to hear, the scale has to have something close to 182¢, 204¢,
> 231¢, 266¢, something to capture the sound of 6:5 and 5:4, something to get 11/8.
> So, what is close to the fewest notes that gives complete 7:8:9:10:11:12 on 8/7, 1/1, 16/9, 8/5, 16/11, 4/3, 8/7, 16/15 in EDO or JI?

22 will give you it, from Orwell or Magic or other things. 31 notes
of Miracle will make it unique. Maybe 58 notes for a unique equal
temperament.

> Also, I'm curious how to generate rank 3 and rank 4 temperaments when I stumble on them.
> For example, what to do with this:
>
> Portent one dimension higher

Yeah, that's a pretty good question. It has something to do with why
the Scala files aren't magically produced on my website.

> Equal Temperament Mappings
> 23571113
> [<31497287107115]
> <416595115142152]
> <4673107129159170]>

What you can do is take two of those, to give a rank temperament
you're familiar with. Then use the other step to define a small
perturbation from it. Or you can generalize distributional evenness
so that the three different step sizes are evenly distributed. There
may be a simple formula for that.

> Reduced Mapping
> 23571113
> [<110357]
> <030-14-5]
> <0010-1-1]>
>
> Generator Tunings (cents)
> [1200.067, 234.186, 2786.754>

Or you can take this (which I believe to be Hermite normal form) and
note that one generator is a slightly detuned octave, and another is a
slightly detuned 5:1. So you can think of the 2-5 plane as just
intonation and use the other generator to give the 3:1. This approach
is probably more straightforward for temperaments where two generators
happen to give you the Pythagorean plane.

> Step Tunings (cents)
> [5.660, 11.068, 12.409>
>
> Tuning Map (cents)
> <1200.067, 1902.624, 2786.754, 3366.014, 4150.323, 4442.783]

The other thing is to design a scale in JI, and apply this tuning to it.

> a generator of 234.186 repeated 41 times makes step sizes of 29.07 and 30.696 if the modulus is 1200 cents.
> (This looks pretty good, actually: 0., 29.07, 58.14, 87.21, 116.28, 146.976, 176.046, 205.116, 234.186, 263.256, 292.326, 321.396, 350.466, 381.162, 410.232, 439.302, 468.372, 497.442, 526.512, 555.582, 584.652, 615.348, 644.418, 673.488, 702.558, 731.628, 760.698, 789.768, 818.838, 849.534, 878.604, 907.674, 936.744, 965.814, 994.884, 1023.954, 1053.024, 1083.72, 1112.79, 1141.86, 1170.93

You're only getting a rank 2 scale there. It may well be a good one
but I can't work out what temperament class it is offhand.

Note: this idea of iterating a generator within a period is called
MOS. Putting the detailed definitions aside, when people talk about
MOS, this is what they mean. Nothing more clever than that. Any MOS
can be interpreted as a rank 2 temperament (or rank 1 in the
degenerate case). Higher rank temperaments will lead to more
complicated scales.

> But I don't understand what to do with the 2786 info (5th partial) that's included under Generator Tunings.
> This is ?Rank 2 because of 29 cent and 30.6 cent step-sizes?

It's rank 2 because it's generated by two intervals, whichever two you take.

To make it rank 3, what you do is stop taking the 5:1 (or 5:4) from
this scale, and make it an independent interval. Maybe with two
keyboards a 5:4 apart.

Graham

🔗Herman Miller <hmiller@...>

🔗caleb morgan <calebmrgn@...>

Slightly tangential, but I'm trying to put together the idea of consistently-fingered chains of 4ths and 5ths and epimorphism from Gene; with reducing the number of notes a little; and hybrid-ism in which the adjacent-note intervals are inconsistent.

This proved trickier than I thought.

But it seems one way to do it is to take and EDO with a fairly accurate 5th, and then
substitute a chain of 5ths and 4ths with consistent spacing, up to the point where the new
5ths and 4ths diverge too much from the EDO to make any sense.

Then substitute freely with pitches that fall between the given locations of 5ths and 4ths, starting with a 5-limit, and adding more when possible. Also, simply leaving the remaining EDO pitches if nothing better strikes one.

! hybrid 27
hybrid 27EDO&5-lim just, with adjusted 4ths & 5ths
27
!0., 0
70.588
88.889, 2
16/15 111
177.778, 4
211.765
282.353
6/5 315
355.556, 8
5/4 389
423.529
494.118
533.333, 12
45/32 590
!
64/45 609
635.294
705.882
776.471
8/5 813
844.444, 19
5/3 884
917.647
988.235
1022.222, 23
15/8 1088.3
1111.111, 25
1129.412
1200 27/0

On Sep 6, 2010, at 5:42 PM, Herman Miller wrote:

> Graham Breed wrote:
> > On 6 September 2010 05:55, caleb morgan <calebmrgn@...> wrote:
>
> >> a generator of 234.186 repeated 41 times makes step sizes of 29.07 and 30.696 if the modulus is 1200 cents.
> >> (This looks pretty good, actually: 0., 29.07, 58.14, 87.21, 116.28, 146.976, 176.046, 205.116, 234.186, 263.256, 292.326, 321.396, 350.466, 381.162, 410.232, 439.302, 468.372, 497.442, 526.512, 555.582, 584.652, 615.348, 644.418, 673.488, 702.558, 731.628, 760.698, 789.768, 818.838, 849.534, 878.604, 907.674, 936.744, 965.814, 994.884, 1023.954, 1053.024, 1083.72, 1112.79, 1141.86, 1170.93
> >
> > You're only getting a rank 2 scale there. It may well be a good one
> > but I can't work out what temperament class it is offhand.
>
> Looks like rodan <1, 1, -1, 3, 6], <0, 3, 17, -1, -13]>.
>
>

🔗caleb morgan <calebmrgn@...>

Oops.

This is a little better.

But I'm still missing a few conceptual bricks from my edifice.

! hybrid 27
hybrid 27EDO&5-lim just, with adjusted 4ths & 5ths
27
!0., !0
70.588
16/15 !2
133.333
177.778 !4
211.765
282.353
6/5 !315
355.556 !8
5/4 !389
423.529
494.118
533.333 !12
45/32 !590 !13
!
64/45 !609 !14
666.667 !15
705.882 !16
755.556 !17
8/5 !813 !18
840.53
5/3 !884
917.647
988.235
1022.222 !23
15/8 !1088.3
1111.111 !25
1129.412
1200 !27/0

On Sep 7, 2010, at 10:16 AM, caleb morgan wrote:

>
> Slightly tangential, but I'm trying to put together the idea of consistently-fingered chains of 4ths and 5ths and epimorphism from Gene; with reducing the number of notes a little; and hybrid-ism in which the adjacent-note intervals are inconsistent.
>
> This proved trickier than I thought.
>
> But it seems one way to do it is to take and EDO with a fairly accurate 5th, and then
> substitute a chain of 5ths and 4ths with consistent spacing, up to the point where the new
> 5ths and 4ths diverge too much from the EDO to make any sense.
>
> Then substitute freely with pitches that fall between the given locations of 5ths and 4ths, starting with a 5-limit, and adding more when possible. Also, simply leaving the remaining EDO pitches if nothing better strikes one.
>
> ! hybrid 27
> hybrid 27EDO&5-lim just, with adjusted 4ths & 5ths
> 27
> !0., 0
> 70.588
> 88.889, 2
> 16/15 111
> 177.778, 4
> 211.765
> 282.353
> 6/5 315
> 355.556, 8
> 5/4 389
> 423.529
> 494.118
> 533.333, 12
> 45/32 590
> !
> 64/45 609
> 635.294
> 705.882
> 776.471
> 8/5 813
> 844.444, 19
> 5/3 884
> 917.647
> 988.235
> 1022.222, 23
> 15/8 1088.3
> 1111.111, 25
> 1129.412
> 1200 27/0
>
> On Sep 6, 2010, at 5:42 PM, Herman Miller wrote:
>
>>
>> Graham Breed wrote:
>> > On 6 September 2010 05:55, caleb morgan <calebmrgn@...> wrote:
>>
>> >> a generator of 234.186 repeated 41 times makes step sizes of 29.07 and 30.696 if the modulus is 1200 cents.
>> >> (This looks pretty good, actually: 0., 29.07, 58.14, 87.21, 116.28, 146.976, 176.046, 205.116, 234.186, 263.256, 292.326, 321.396, 350.466, 381.162, 410.232, 439.302, 468.372, 497.442, 526.512, 555.582, 584.652, 615.348, 644.418, 673.488, 702.558, 731.628, 760.698, 789.768, 818.838, 849.534, 878.604, 907.674, 936.744, 965.814, 994.884, 1023.954, 1053.024, 1083.72, 1112.79, 1141.86, 1170.93
>> >
>> > You're only getting a rank 2 scale there. It may well be a good one
>> > but I can't work out what temperament class it is offhand.
>>
>> Looks like rodan <1, 1, -1, 3, 6], <0, 3, 17, -1, -13]>.
>>
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

Oh, there's interest!

What I'm hoping to achieve, I guess, is to understand some rules of thumb about when it's possible and how to do it myself.

Are there certain numbers of pitches per octave where consistency is impossible?

(I'm sort of afraid that the answer will be over my head.)

Meanwhile, the list of scales with which I hope to write pieces has grown to nearly 50, with some clear favorites--GWS 46-notes epimorphic 13-limit, GWS newts, 46EDO, 41 EDO, any some of my own JI scales.

Plus, just to find out some stuff by trial-and-error myself, I'm looking into equal divisions of intervals other than the octave.

But "Dwarf27" looks really good.

In terms of my overall project, I was sort of hoping to accumulate a bunch of really good tunings, then switch to concentrating on composing with each one.

It's fun to make new scales, but I can't hope to approach the kind of mastery that I see here from you and a few others, so I do intend to change my focus back to composing at some point.

caleb

On Sep 8, 2010, at 11:23 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > I suppose that a completely consistent fingering (5ths all the same number of steps) is mathematically impossible with a 5th at 703.06613 with 27 steps?
>
> Below is something a little closer to what you want, with fifths mostly in interval class 16 but some, including one good one, in 15. I'll return to this question if there is interest.
>
> ! dwarf27_7tempered.scl
> !
> Irregularly tempered Dwarf(<27 43 63 76|)
> 27
> !
> 8.50411
> 44.05819
> 94.02735
> 155.44995
> 204.62445
> 239.75461
> 275.21962
> 300.57603
> 386.44020
> 410.42948
> 471.11121
> 506.30767
> 541.94592
> 591.15701
> 616.87618
> 702.25982
> 737.66181
> 772.99566
> 797.33536
> 857.59196
> 907.39146
> 969.00623
> 977.50974
> 1003.78401
> 1088.74607
> 1113.63615
> 1200.00000
>
>

🔗caleb morgan <calebmrgn@...>

Caleb plays "27Dwarf" :

Hmm: 5ths are all always 16 steps, but some are a long way from 3:2 702 cents.

4ths: 12 steps, except 16:9 to 32:27 approx is 11 steps, and the 4ths are charmingly different-sized-sounding.

pattern of 12-12-11 steps on the 4ths?

good 21/8 (7th par of 3)

Those are my first impressions.

caleb

On Sep 9, 2010, at 5:36 AM, caleb morgan wrote:

>
> Oh, there's interest!
>
> What I'm hoping to achieve, I guess, is to understand some rules of thumb about when it's possible and how to do it myself.
>
> Are there certain numbers of pitches per octave where consistency is impossible?
>
> (I'm sort of afraid that the answer will be over my head.)
>
> Meanwhile, the list of scales with which I hope to write pieces has grown to nearly 50, with some clear favorites--GWS 46-notes epimorphic 13-limit, GWS newts, 46EDO, 41 EDO, any some of my own JI scales.
>
> Plus, just to find out some stuff by trial-and-error myself, I'm looking into equal divisions of intervals other than the octave.
>
> But "Dwarf27" looks really good.
>
> In terms of my overall project, I was sort of hoping to accumulate a bunch of really good tunings, then switch to concentrating on composing with each one.
>
> It's fun to make new scales, but I can't hope to approach the kind of mastery that I see here from you and a few others, so I do intend to change my focus back to composing at some point.
>
> caleb
>
>
>
> On Sep 8, 2010, at 11:23 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>
>> > I suppose that a completely consistent fingering (5ths all the same number of steps) is mathematically impossible with a 5th at 703.06613 with 27 steps?
>>
>> Below is something a little closer to what you want, with fifths mostly in interval class 16 but some, including one good one, in 15. I'll return to this question if there is interest.
>>
>> ! dwarf27_7tempered.scl
>> !
>> Irregularly tempered Dwarf(<27 43 63 76|)
>> 27
>> !
>> 8.50411
>> 44.05819
>> 94.02735
>> 155.44995
>> 204.62445
>> 239.75461
>> 275.21962
>> 300.57603
>> 386.44020
>> 410.42948
>> 471.11121
>> 506.30767
>> 541.94592
>> 591.15701
>> 616.87618
>> 702.25982
>> 737.66181
>> 772.99566
>> 797.33536
>> 857.59196
>> 907.39146
>> 969.00623
>> 977.50974
>> 1003.78401
>> 1088.74607
>> 1113.63615
>> 1200.00000
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

Here's a new 42-note scale (14x3) by me, called "Beast Augmentation".

5/4 (386.3315 cents) is divided into 23 parts, certain pretty tones are selected.

Then that pattern is repeated at 400 and 800 cents.

It's close to 11-limit JI in some ways, but has more symmetry and regularity.

!Beast Augmentation by Caleb
5/4 in 23 parts, culled, repeated @ 400 and 800
42
!0.,
33.592,
67.185,
100.777,
134.37,
151.166,
167.962,
184.759,
201.555,
235.147,
268.74,
302.332,
319.129,
386.315
!
400
433.592
467.185
500.777
534.37
551.166
567.962
584.759
601.555
635.147
668.74
702.332
719.129
786.315
!
800
833.592
867.185
900.777
934.37
951.166
967.962
984.759
1001.555
1035.147
1068.74
1102.332
1119.129
1186.315
!
1200

On Sep 9, 2010, at 6:27 AM, caleb morgan wrote:

> Caleb plays "27Dwarf" :
>
>
> Hmm: 5ths are all always 16 steps, but some are a long way from 3:2 702 cents.
>
> 4ths: 12 steps, except 16:9 to 32:27 approx is 11 steps, and the 4ths are charmingly different-sized-sounding.
>
> pattern of 12-12-11 steps on the 4ths?
>
> good 21/8 (7th par of 3)
>
> Those are my first impressions.
>
> caleb
>
>
>
>
>
> On Sep 9, 2010, at 5:36 AM, caleb morgan wrote:
>
>>
>>
>> Oh, there's interest!
>>
>> What I'm hoping to achieve, I guess, is to understand some rules of thumb about when it's possible and how to do it myself.
>>
>> Are there certain numbers of pitches per octave where consistency is impossible?
>>
>> (I'm sort of afraid that the answer will be over my head.)
>>
>> Meanwhile, the list of scales with which I hope to write pieces has grown to nearly 50, with some clear favorites--GWS 46-notes epimorphic 13-limit, GWS newts, 46EDO, 41 EDO, any some of my own JI scales.
>>
>> Plus, just to find out some stuff by trial-and-error myself, I'm looking into equal divisions of intervals other than the octave.
>>
>> But "Dwarf27" looks really good.
>>
>> In terms of my overall project, I was sort of hoping to accumulate a bunch of really good tunings, then switch to concentrating on composing with each one.
>>
>> It's fun to make new scales, but I can't hope to approach the kind of mastery that I see here from you and a few others, so I do intend to change my focus back to composing at some point.
>>
>> caleb
>>
>>
>>
>> On Sep 8, 2010, at 11:23 PM, genewardsmith wrote:
>>
>>>
>>>
>>>
>>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>>
>>> > I suppose that a completely consistent fingering (5ths all the same number of steps) is mathematically impossible with a 5th at 703.06613 with 27 steps?
>>>
>>> Below is something a little closer to what you want, with fifths mostly in interval class 16 but some, including one good one, in 15. I'll return to this question if there is interest.
>>>
>>> ! dwarf27_7tempered.scl
>>> !
>>> Irregularly tempered Dwarf(<27 43 63 76|)
>>> 27
>>> !
>>> 8.50411
>>> 44.05819
>>> 94.02735
>>> 155.44995
>>> 204.62445
>>> 239.75461
>>> 275.21962
>>> 300.57603
>>> 386.44020
>>> 410.42948
>>> 471.11121
>>> 506.30767
>>> 541.94592
>>> 591.15701
>>> 616.87618
>>> 702.25982
>>> 737.66181
>>> 772.99566
>>> 797.33536
>>> 857.59196
>>> 907.39146
>>> 969.00623
>>> 977.50974
>>> 1003.78401
>>> 1088.74607
>>> 1113.63615
>>> 1200.00000
>>>
>>
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

Sorry, same thing cleaned up for better Scala format:

!Beast Augmentation by Caleb
5/4 in 23 parts, culled, repeated @ 400 and 800
42
!0.
33.592
67.185
100.777
134.37
151.166
167.962
184.759
201.555
235.147
268.74
302.332
319.129
386.315
!
400.000
433.592
467.185
500.777
534.37
551.166
567.962
584.759
601.555
635.147
668.74
702.332
719.129
786.315
!
800.000
833.592
867.185
900.777
934.37
951.166
967.962
984.759
1001.555
1035.147
1068.74
1102.332
1119.129
1186.315
!
1200.000

On Sep 9, 2010, at 9:05 AM, caleb morgan wrote:

>
>
> Here's a new 42-note scale (14x3) by me, called "Beast Augmentation".
>
> 5/4 (386.3315 cents) is divided into 23 parts, certain pretty tones are selected.
>
> Then that pattern is repeated at 400 and 800 cents.
>
> It's close to 11-limit JI in some ways, but has more symmetry and regularity.
>
> !Beast Augmentation by Caleb
> 5/4 in 23 parts, culled, repeated @ 400 and 800
> 42
> !0.,
> 33.592,
> 67.185,
> 100.777,
> 134.37,
> 151.166,
> 167.962,
> 184.759,
> 201.555,
> 235.147,
> 268.74,
> 302.332,
> 319.129,
> 386.315
> !
> 400
> 433.592
> 467.185
> 500.777
> 534.37
> 551.166
> 567.962
> 584.759
> 601.555
> 635.147
> 668.74
> 702.332
> 719.129
> 786.315
> !
> 800
> 833.592
> 867.185
> 900.777
> 934.37
> 951.166
> 967.962
> 984.759
> 1001.555
> 1035.147
> 1068.74
> 1102.332
> 1119.129
> 1186.315
> !
> 1200
>
>
>
>
>
>
>
>
>
>
> On Sep 9, 2010, at 6:27 AM, caleb morgan wrote:
>
>>
>> Caleb plays "27Dwarf" :
>>
>>
>> Hmm: 5ths are all always 16 steps, but some are a long way from 3:2 702 cents.
>>
>> 4ths: 12 steps, except 16:9 to 32:27 approx is 11 steps, and the 4ths are charmingly different-sized-sounding.
>>
>> pattern of 12-12-11 steps on the 4ths?
>>
>> good 21/8 (7th par of 3)
>>
>> Those are my first impressions.
>>
>> caleb
>>
>>
>>
>>
>>
>> On Sep 9, 2010, at 5:36 AM, caleb morgan wrote:
>>
>>>
>>>
>>> Oh, there's interest!
>>>
>>> What I'm hoping to achieve, I guess, is to understand some rules of thumb about when it's possible and how to do it myself.
>>>
>>> Are there certain numbers of pitches per octave where consistency is impossible?
>>>
>>> (I'm sort of afraid that the answer will be over my head.)
>>>
>>> Meanwhile, the list of scales with which I hope to write pieces has grown to nearly 50, with some clear favorites--GWS 46-notes epimorphic 13-limit, GWS newts, 46EDO, 41 EDO, any some of my own JI scales.
>>>
>>> Plus, just to find out some stuff by trial-and-error myself, I'm looking into equal divisions of intervals other than the octave.
>>>
>>> But "Dwarf27" looks really good.
>>>
>>> In terms of my overall project, I was sort of hoping to accumulate a bunch of really good tunings, then switch to concentrating on composing with each one.
>>>
>>> It's fun to make new scales, but I can't hope to approach the kind of mastery that I see here from you and a few others, so I do intend to change my focus back to composing at some point.
>>>
>>> caleb
>>>
>>>
>>>
>>> On Sep 8, 2010, at 11:23 PM, genewardsmith wrote:
>>>
>>>>
>>>>
>>>>
>>>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>>>
>>>> > I suppose that a completely consistent fingering (5ths all the same number of steps) is mathematically impossible with a 5th at 703.06613 with 27 steps?
>>>>
>>>> Below is something a little closer to what you want, with fifths mostly in interval class 16 but some, including one good one, in 15. I'll return to this question if there is interest.
>>>>
>>>> ! dwarf27_7tempered.scl
>>>> !
>>>> Irregularly tempered Dwarf(<27 43 63 76|)
>>>> 27
>>>> !
>>>> 8.50411
>>>> 44.05819
>>>> 94.02735
>>>> 155.44995
>>>> 204.62445
>>>> 239.75461
>>>> 275.21962
>>>> 300.57603
>>>> 386.44020
>>>> 410.42948
>>>> 471.11121
>>>> 506.30767
>>>> 541.94592
>>>> 591.15701
>>>> 616.87618
>>>> 702.25982
>>>> 737.66181
>>>> 772.99566
>>>> 797.33536
>>>> 857.59196
>>>> 907.39146
>>>> 969.00623
>>>> 977.50974
>>>> 1003.78401
>>>> 1088.74607
>>>> 1113.63615
>>>> 1200.00000
>>>>
>>>
>>>
>>
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

Here's a 48-tone version of the same thing.

This combines an attempt at epimorphism with 48-tone fingering with 11 and 13-limit JI with a concession to the standard keyboard pattern of 12 notes.

It's pretty learnable on a standard keyboard, because of the 48-note pattern and consistent 5ths.

It beats slightly differently than 72EDO or 48 EDO because of the slight stretch, making the major 3rd and 5th nearly perfect.

The formula was, again, to divide 5/4 into 23 equal parts and repeat those pitches at 400 and 800 cents for something close to 69EDO, and then to throw away some pitches, keeping 4 pitches to every approximate 100 cents, repeating pitches starting at 702.33 so that they were accurate 5ths above the starting pitches.

Caleb

!Beast Augmentation by Caleb
5/4 in 23 parts, repeated @ 400 and 800
48
!0.,
50.389
67.185
83.981
!
117.574
151.166
167.962
184.759
!
201.555
235.147
268.74
285.536
!
302.332
319.129
352.721
386.315
!
400
416.796
433.592
450.389
!
500.777
551.166
567.962
584.759
!
601.555
618.351
635.147
651.944
!
702.332
752.721
769.517
786.315
!
816.796
850.389
867.185
883.981
!
900.777
934.37
967.962
984.759
!
1001.555
1018.351
1051.944
1085.536
!
1102.332
1119.129
1135.925
1152.721
!
1200

On Sep 9, 2010, at 9:27 AM, caleb morgan wrote:

>
> Sorry, same thing cleaned up for better Scala format:
>
> !Beast Augmentation by Caleb
> 5/4 in 23 parts, culled, repeated @ 400 and 800
> 42
> !0.
> 33.592
> 67.185
> 100.777
> 134.37
> 151.166
> 167.962
> 184.759
> 201.555
> 235.147
> 268.74
> 302.332
> 319.129
> 386.315
> !
> 400.000
> 433.592
> 467.185
> 500.777
> 534.37
> 551.166
> 567.962
> 584.759
> 601.555
> 635.147
> 668.74
> 702.332
> 719.129
> 786.315
> !
> 800.000
> 833.592
> 867.185
> 900.777
> 934.37
> 951.166
> 967.962
> 984.759
> 1001.555
> 1035.147
> 1068.74
> 1102.332
> 1119.129
> 1186.315
> !
> 1200.000
>
>
>
>
> On Sep 9, 2010, at 9:05 AM, caleb morgan wrote:
>
>>
>>
>>
>> Here's a new 42-note scale (14x3) by me, called "Beast Augmentation".
>>
>> 5/4 (386.3315 cents) is divided into 23 parts, certain pretty tones are selected.
>>
>> Then that pattern is repeated at 400 and 800 cents.
>>
>> It's close to 11-limit JI in some ways, but has more symmetry and regularity.
>>
>> !Beast Augmentation by Caleb
>> 5/4 in 23 parts, culled, repeated @ 400 and 800
>> 42
>> !0.,
>> 33.592,
>> 67.185,
>> 100.777,
>> 134.37,
>> 151.166,
>> 167.962,
>> 184.759,
>> 201.555,
>> 235.147,
>> 268.74,
>> 302.332,
>> 319.129,
>> 386.315
>> !
>> 400
>> 433.592
>> 467.185
>> 500.777
>> 534.37
>> 551.166
>> 567.962
>> 584.759
>> 601.555
>> 635.147
>> 668.74
>> 702.332
>> 719.129
>> 786.315
>> !
>> 800
>> 833.592
>> 867.185
>> 900.777
>> 934.37
>> 951.166
>> 967.962
>> 984.759
>> 1001.555
>> 1035.147
>> 1068.74
>> 1102.332
>> 1119.129
>> 1186.315
>> !
>> 1200
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> On Sep 9, 2010, at 6:27 AM, caleb morgan wrote:
>>
>>>
>>> Caleb plays "27Dwarf" :
>>>
>>>
>>> Hmm: 5ths are all always 16 steps, but some are a long way from 3:2 702 cents.
>>>
>>> 4ths: 12 steps, except 16:9 to 32:27 approx is 11 steps, and the 4ths are charmingly different-sized-sounding.
>>>
>>> pattern of 12-12-11 steps on the 4ths?
>>>
>>> good 21/8 (7th par of 3)
>>>
>>> Those are my first impressions.
>>>
>>> caleb
>>>
>>>
>>>
>>>
>>>
>>> On Sep 9, 2010, at 5:36 AM, caleb morgan wrote:
>>>
>>>>
>>>>
>>>> Oh, there's interest!
>>>>
>>>> What I'm hoping to achieve, I guess, is to understand some rules of thumb about when it's possible and how to do it myself.
>>>>
>>>> Are there certain numbers of pitches per octave where consistency is impossible?
>>>>
>>>> (I'm sort of afraid that the answer will be over my head.)
>>>>
>>>> Meanwhile, the list of scales with which I hope to write pieces has grown to nearly 50, with some clear favorites--GWS 46-notes epimorphic 13-limit, GWS newts, 46EDO, 41 EDO, any some of my own JI scales.
>>>>
>>>> Plus, just to find out some stuff by trial-and-error myself, I'm looking into equal divisions of intervals other than the octave.
>>>>
>>>> But "Dwarf27" looks really good.
>>>>
>>>> In terms of my overall project, I was sort of hoping to accumulate a bunch of really good tunings, then switch to concentrating on composing with each one.
>>>>
>>>> It's fun to make new scales, but I can't hope to approach the kind of mastery that I see here from you and a few others, so I do intend to change my focus back to composing at some point.
>>>>
>>>> caleb
>>>>
>>>>
>>>>
>>>> On Sep 8, 2010, at 11:23 PM, genewardsmith wrote:
>>>>
>>>>>
>>>>>
>>>>>
>>>>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>>>>
>>>>> > I suppose that a completely consistent fingering (5ths all the same number of steps) is mathematically impossible with a 5th at 703.06613 with 27 steps?
>>>>>
>>>>> Below is something a little closer to what you want, with fifths mostly in interval class 16 but some, including one good one, in 15. I'll return to this question if there is interest.
>>>>>
>>>>> ! dwarf27_7tempered.scl
>>>>> !
>>>>> Irregularly tempered Dwarf(<27 43 63 76|)
>>>>> 27
>>>>> !
>>>>> 8.50411
>>>>> 44.05819
>>>>> 94.02735
>>>>> 155.44995
>>>>> 204.62445
>>>>> 239.75461
>>>>> 275.21962
>>>>> 300.57603
>>>>> 386.44020
>>>>> 410.42948
>>>>> 471.11121
>>>>> 506.30767
>>>>> 541.94592
>>>>> 591.15701
>>>>> 616.87618
>>>>> 702.25982
>>>>> 737.66181
>>>>> 772.99566
>>>>> 797.33536
>>>>> 857.59196
>>>>> 907.39146
>>>>> 969.00623
>>>>> 977.50974
>>>>> 1003.78401
>>>>> 1088.74607
>>>>> 1113.63615
>>>>> 1200.00000
>>>>>
>>>>
>>>>
>>>
>>>
>>
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

P.S. to Gene, or any one else who is interested, I'd be very grateful for 48-pitch systems that are compromises between JI and EDO, as I have attempted in my previous post.

Probably Gene could do it better, with some clever temperings and additions.

caleb

On Sep 9, 2010, at 9:27 AM, caleb morgan wrote:

🔗anton_pann <anton_pann@...>

🔗caleb morgan <calebmrgn@...>

Well, there's always this--though it didn't take much ingenuity:

The 48 Boring & Pierce

!Roaring, Boring, and Piercing
48 Boring & Pierce
48
!0.
33.
50.
66.
!
100.
133.
150.
167.
!
200.
233.
250.
267.
!
300.
333.
350.
367.
!
400.
433.
450.
467.
!
500.
533.
550.
567.
!
600.
633.
650.
667.
!
700.
733.
750.
767.
!
800.
833.
850.
867.
!
900.
933.
950.
967.
!
1000.
1033.
1050.
1067.
!
1100.
1133.
1150.
1167.
!
1200.

On Sep 9, 2010, at 12:08 PM, caleb morgan wrote:

> P.S. to Gene, or any one else who is interested, I'd be very grateful for 48-pitch systems that are compromises between JI and EDO, as I have attempted in my previous post.
>
>
> Probably Gene could do it better, with some clever temperings and additions.
>
> caleb
>
>
>
>
>
> On Sep 9, 2010, at 9:27 AM, caleb morgan wrote:
>
>>
>>
>> Sorry, same thing cleaned up for better Scala format:
>>
>> !Beast Augmentation by Caleb
>> 5/4 in 23 parts, culled, repeated @ 400 and 800
>> 42
>> !0.
>> 33.592
>> 67.185
>> 100.777
>> 134.37
>> 151.166
>> 167.962
>> 184.759
>> 201.555
>> 235.147
>> 268.74
>> 302.332
>> 319.129
>> 386.315
>> !
>> 400.000
>> 433.592
>> 467.185
>> 500.777
>> 534.37
>> 551.166
>> 567.962
>> 584.759
>> 601.555
>> 635.147
>> 668.74
>> 702.332
>> 719.129
>> 786.315
>> !
>> 800.000
>> 833.592
>> 867.185
>> 900.777
>> 934.37
>> 951.166
>> 967.962
>> 984.759
>> 1001.555
>> 1035.147
>> 1068.74
>> 1102.332
>> 1119.129
>> 1186.315
>> !
>> 1200.000
>>
>>
>>
>>
>> On Sep 9, 2010, at 9:05 AM, caleb morgan wrote:
>>
>>>
>>>
>>>
>>> Here's a new 42-note scale (14x3) by me, called "Beast Augmentation".
>>>
>>> 5/4 (386.3315 cents) is divided into 23 parts, certain pretty tones are selected.
>>>
>>> Then that pattern is repeated at 400 and 800 cents.
>>>
>>> It's close to 11-limit JI in some ways, but has more symmetry and regularity.
>>>
>>> !Beast Augmentation by Caleb
>>> 5/4 in 23 parts, culled, repeated @ 400 and 800
>>> 42
>>> !0.,
>>> 33.592,
>>> 67.185,
>>> 100.777,
>>> 134.37,
>>> 151.166,
>>> 167.962,
>>> 184.759,
>>> 201.555,
>>> 235.147,
>>> 268.74,
>>> 302.332,
>>> 319.129,
>>> 386.315
>>> !
>>> 400
>>> 433.592
>>> 467.185
>>> 500.777
>>> 534.37
>>> 551.166
>>> 567.962
>>> 584.759
>>> 601.555
>>> 635.147
>>> 668.74
>>> 702.332
>>> 719.129
>>> 786.315
>>> !
>>> 800
>>> 833.592
>>> 867.185
>>> 900.777
>>> 934.37
>>> 951.166
>>> 967.962
>>> 984.759
>>> 1001.555
>>> 1035.147
>>> 1068.74
>>> 1102.332
>>> 1119.129
>>> 1186.315
>>> !
>>> 1200
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> On Sep 9, 2010, at 6:27 AM, caleb morgan wrote:
>>>
>>>>
>>>> Caleb plays "27Dwarf" :
>>>>
>>>>
>>>> Hmm: 5ths are all always 16 steps, but some are a long way from 3:2 702 cents.
>>>>
>>>> 4ths: 12 steps, except 16:9 to 32:27 approx is 11 steps, and the 4ths are charmingly different-sized-sounding.
>>>>
>>>> pattern of 12-12-11 steps on the 4ths?
>>>>
>>>> good 21/8 (7th par of 3)
>>>>
>>>> Those are my first impressions.
>>>>
>>>> caleb
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> On Sep 9, 2010, at 5:36 AM, caleb morgan wrote:
>>>>
>>>>>
>>>>>
>>>>> Oh, there's interest!
>>>>>
>>>>> What I'm hoping to achieve, I guess, is to understand some rules of thumb about when it's possible and how to do it myself.
>>>>>
>>>>> Are there certain numbers of pitches per octave where consistency is impossible?
>>>>>
>>>>> (I'm sort of afraid that the answer will be over my head.)
>>>>>
>>>>> Meanwhile, the list of scales with which I hope to write pieces has grown to nearly 50, with some clear favorites--GWS 46-notes epimorphic 13-limit, GWS newts, 46EDO, 41 EDO, any some of my own JI scales.
>>>>>
>>>>> Plus, just to find out some stuff by trial-and-error myself, I'm looking into equal divisions of intervals other than the octave.
>>>>>
>>>>> But "Dwarf27" looks really good.
>>>>>
>>>>> In terms of my overall project, I was sort of hoping to accumulate a bunch of really good tunings, then switch to concentrating on composing with each one.
>>>>>
>>>>> It's fun to make new scales, but I can't hope to approach the kind of mastery that I see here from you and a few others, so I do intend to change my focus back to composing at some point.
>>>>>
>>>>> caleb
>>>>>
>>>>>
>>>>>
>>>>> On Sep 8, 2010, at 11:23 PM, genewardsmith wrote:
>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>>>>>
>>>>>> > I suppose that a completely consistent fingering (5ths all the same number of steps) is mathematically impossible with a 5th at 703.06613 with 27 steps?
>>>>>>
>>>>>> Below is something a little closer to what you want, with fifths mostly in interval class 16 but some, including one good one, in 15. I'll return to this question if there is interest.
>>>>>>
>>>>>> ! dwarf27_7tempered.scl
>>>>>> !
>>>>>> Irregularly tempered Dwarf(<27 43 63 76|)
>>>>>> 27
>>>>>> !
>>>>>> 8.50411
>>>>>> 44.05819
>>>>>> 94.02735
>>>>>> 155.44995
>>>>>> 204.62445
>>>>>> 239.75461
>>>>>> 275.21962
>>>>>> 300.57603
>>>>>> 386.44020
>>>>>> 410.42948
>>>>>> 471.11121
>>>>>> 506.30767
>>>>>> 541.94592
>>>>>> 591.15701
>>>>>> 616.87618
>>>>>> 702.25982
>>>>>> 737.66181
>>>>>> 772.99566
>>>>>> 797.33536
>>>>>> 857.59196
>>>>>> 907.39146
>>>>>> 969.00623
>>>>>> 977.50974
>>>>>> 1003.78401
>>>>>> 1088.74607
>>>>>> 1113.63615
>>>>>> 1200.00000
>>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>
>>>
>>
>>
>
>
>

🔗Mike Battaglia <battaglia01@...>

🔗caleb morgan <calebmrgn@...>

Well, what I lack in tuning acumen I make up in ridiculous titles.

Here's one that works in 36 or 48-notes, a generator of 352.721, stretched octave (modulus) of 1203.468. (That is, 352.721 is multiplied by every N Mod 1203.468 up to 36 or 48.)

But I actually don't like it as much as Beast Aug.

caleb

!48 tones of 352.721
352.721x48
48
0.,
21.083
41.028
62.111
83.194
124.222
145.305
166.388
186.333
207.416
228.499
269.527
290.61
311.693
331.638
352.721
373.804
414.832
435.915
476.943
498.026
519.109
539.054
560.137
581.22
622.248
643.331
664.414
684.359
705.442
726.525
767.553
788.636
829.664
850.747
871.83
891.775
912.858
933.941
974.969
996.052
1017.135
1037.08
1058.163
1079.246
1120.274
1141.357
1182.385
1203.468

On Sep 9, 2010, at 3:35 PM, Mike Battaglia wrote:

> On Thu, Sep 9, 2010 at 9:05 AM, caleb morgan <calebmrgn@...> wrote:
> >
> > Here's a new 42-note scale (14x3) by me, called "Beast Augmentation".
>
> Hahahaha!
>
> > 5/4 (386.3315 cents) is divided into 23 parts, certain pretty tones are selected.
> > Then that pattern is repeated at 400 and 800 cents.
> > It's close to 11-limit JI in some ways, but has more symmetry and regularity.
> > !Beast Augmentation by Caleb
> > 5/4 in 23 parts, culled, repeated @ 400 and 800
> > 42
> > !0.,
> > 33.592,
> > 67.185,
> > 100.777,
> > 134.37,
> > 151.166,
> > 167.962,
> > 184.759,
> > 201.555,
> > 235.147,
> > 268.74,
> > 302.332,
> > 319.129,
> > 386.315
> > !
> > 400
> > 433.592
> > 467.185
> > 500.777
> > 534.37
> > 551.166
> > 567.962
> > 584.759
> > 601.555
> > 635.147
> > 668.74
> > 702.332
> > 719.129
> > 786.315
> > !
> > 800
> > 833.592
> > 867.185
> > 900.777
> > 934.37
> > 951.166
> > 967.962
> > 984.759
> > 1001.555
> > 1035.147
> > 1068.74
> > 1102.332
> > 1119.129
> > 1186.315
> > !
> > 1200
>
> Looks like perhaps you're ready to take the plunge into 72-tet?
> Probably the easiest temperament to think in ever.
>
> -Mike
>

🔗genewardsmith <genewardsmith@...>

🔗robert <robertthomasmartin@...>

🔗caleb morgan <calebmrgn@...>

I'm sorry for excessive posting if this has been the case.

I'm in a scale-design phase, which I hope to wrap up shortly.

It's been a process of learning what's possible, what's desirable, what's convenient, and what's easily learnable.

I've been circling 'round something. It involves JI/EDO hybrids of between 36 and 48 pitches.

43 pitches turns out to fit two "octaves" perfectly, so it seems optimal for a number of reasons.

A standard keyboard is a given.

So, bear with me if I change my specs slightly one more time.

The "perfect" scale would be:

Size: 43 notes

Consistency: Not *entirely* consistent in smallest-step sizes, but fingering of closest equivalents to 3/2, 4/3, 9/8, 16/9, 5/4, 8/5, etc. would always be the *same* number of key-steps.

5ths: wide but no wider than 704.35

5/4's: wide by no more than 6 cents or dead on

8/7, 7/6, 12/7, 7/4 off by no more than 6 cents

octave: within 4 cents, preferably wide, say 1204.

Consistency in every "key"*: *not* necessary. However, every "key" that has 1/1 as a member must be fairly in-tune, that is, have a good 3:4:5 triad.

There can be a number of "bad" keys, such as the one on the "second" degree of scale of 43.

There can be a very few "filler" notes for the sole purpose of making the fingering consistent. You might never use them.

The "error in the system" should be greatest around the keys that you wouldn't use as being harmonically-related to 1/1. So, /8, /9 etc to /11, perhaps even /13 would be accurate enough.

So would the "strange" tonalities on 9/8, and 5/4, and 27/16 and 5/3, even.

But some "keys" could um, suck, to put it bluntly.

So, it's possible to play this scale with good consonant 5:6:7:8:9:10:11 in at least 7 keys, with many more good but non OT scales available on other keys.

I think this is do-able. I think it's important, even.

*Some* beating in the 5's, 9's, 11's 13's, and 15's is even desirable, but not too much!

It may already exist.

It might look like a slight tempering of Partch's scale.

This is my current thinking.

I'm quite serious.

It would be great to find one scale that is better than all the rest for a standard keyboard, allowing some modulation, giving good 7 and 11-limit harmony, with chains of 4ths and 5ths around 7-10 notes long on either side. They don't have to be the same size, but they do have to finger the same, and be acceptable, not wolves or voles.

It would be great to be able to settle down with one "best" scale. This is as close as I can get.

This is the best I can do to be serious, accessible, educational, respectful, and blunt, yet speak in the closest equivalent to ordinary language, free of jargon.

Caleb

On Sep 9, 2010, at 4:14 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
> >
> > Here's a 48-tone version of the same thing.
> >
> > This combines an attempt at epimorphism with 48-tone fingering with 11 and 13-limit JI with a concession to the standard keyboard pattern of 12 notes.
> >
> > It's pretty learnable on a standard keyboard, because of the 48-note pattern and consistent 5ths.
>
> This should be easily learnable, and has a lot of nice 11-limit harmony:
>
> ! compton48.scl
> !
> Compton[48] 11-limit tweaked
> 48
> !
> 50.02948
> 66.62693
> 83.40254
> 100.00000
> 150.02948
> 166.62693
> 183.40254
> 200.00000
> 250.02948
> 266.62693
> 283.40254
> 300.00000
> 350.02948
> 366.62693
> 383.40254
> 400.00000
> 450.02948
> 466.62693
> 483.40254
> 500.00000
> 550.02948
> 566.62693
> 583.40254
> 600.00000
> 650.02948
> 666.62693
> 683.40254
> 700.00000
> 750.02948
> 766.62693
> 783.40254
> 800.00000
> 850.02948
> 866.62693
> 883.40254
> 900.00000
> 950.02948
> 966.62693
> 983.40254
> 1000.00000
> 1050.02948
> 1066.62693
> 1083.40254
> 1100.00000
> 1150.02948
> 1166.62693
> 1183.40254
> 1200.00000
>
>

🔗caleb morgan <calebmrgn@...>

In (what I hope is) the end, the best I can do is the scale by Gene Ward Smith,
minus the two notes nearest 1/1 on the bottom, and near 2/1 on the top.

I'll just learn to love the inconsistent fingering, and if I need other tonal relationships, I'll "modulate" by pitch-bending the whole tuning down to some other level.

We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.
Through the unknown, unremembered gate
When the last of earth left to discover
Is that which was the beginning;
At the source of the longest river
The voice of the hidden waterfall
And the children in the apple-tree
Not known, because not looked for
But heard, half-heard, in the stillness
Between two waves of the sea.
Quick now, here, now, always—
A condition of complete simplicity
(Costing not less than everything)
And all shall be well and
All manner of thing shall be well
When the tongues of flames are in-folded
Into the crowned knot of fire
And the fire and the rose are one.

! cal42.scl
42-n 4 Cal by G.W.Smith "required"
42
!
80.6369
104.0612
126.6171
160.4026
184.7357
208.1710
231.9308
263.2192
288.7711
!
311.2289
336.7808
368.0692
391.8290
415.2643
439.5974
473.3829
495.9388
519.3631
543.7424
577.0749
600.0000
!
!
622.9251
656.2576
680.6369
704.0612
726.6171
760.4026
784.7357
808.1710
831.9308
863.2192
888.7711
911.2289
!
936.7808
968.0692
991.8290
1015.2643
1039.5974
1073.3829
1095.9388
1119.3631
1200.0000

Just for yucks:

On Sep 10, 2010, at 9:12 AM, caleb morgan wrote:

>
> I'm sorry for excessive posting if this has been the case.
>
> I'm in a scale-design phase, which I hope to wrap up shortly.
>
> It's been a process of learning what's possible, what's desirable, what's convenient, and what's easily learnable.
>
> I've been circling 'round something. It involves JI/EDO hybrids of between 36 and 48 pitches.
>
> 43 pitches turns out to fit two "octaves" perfectly, so it seems optimal for a number of reasons.
>
> A standard keyboard is a given.
>
> So, bear with me if I change my specs slightly one more time.
>
> The "perfect" scale would be:
>
> Size: 43 notes
>
> Consistency: Not *entirely* consistent in smallest-step sizes, but fingering of closest equivalents to 3/2, 4/3, 9/8, 16/9, 5/4, 8/5, etc. would always be the *same* number of key-steps.
>
> 5ths: wide but no wider than 704.35
>
> 5/4's: wide by no more than 6 cents or dead on
>
> 8/7, 7/6, 12/7, 7/4 off by no more than 6 cents
>
> octave: within 4 cents, preferably wide, say 1204.
>
> Consistency in every "key"*: *not* necessary. However, every "key" that has 1/1 as a member must be fairly in-tune, that is, have a good 3:4:5 triad.
>
> There can be a number of "bad" keys, such as the one on the "second" degree of scale of 43.
>
> There can be a very few "filler" notes for the sole purpose of making the fingering consistent. You might never use them.
>
> The "error in the system" should be greatest around the keys that you wouldn't use as being harmonically-related to 1/1. So, /8, /9 etc to /11, perhaps even /13 would be accurate enough.
>
> So would the "strange" tonalities on 9/8, and 5/4, and 27/16 and 5/3, even.
>
> But some "keys" could um, suck, to put it bluntly.
>
> So, it's possible to play this scale with good consonant 5:6:7:8:9:10:11 in at least 7 keys, with many more good but non OT scales available on other keys.
>
> I think this is do-able. I think it's important, even.
>
> *Some* beating in the 5's, 9's, 11's 13's, and 15's is even desirable, but not too much!
>
> It may already exist.
>
> It might look like a slight tempering of Partch's scale.
>
> This is my current thinking.
>
> I'm quite serious.
>
> It would be great to find one scale that is better than all the rest for a standard keyboard, allowing some modulation, giving good 7 and 11-limit harmony, with chains of 4ths and 5ths around 7-10 notes long on either side. They don't have to be the same size, but they do have to finger the same, and be acceptable, not wolves or voles.
>
> It would be great to be able to settle down with one "best" scale. This is as close as I can get.
>
> This is the best I can do to be serious, accessible, educational, respectful, and blunt, yet speak in the closest equivalent to ordinary language, free of jargon.
>
> Caleb
>
>
>
>
>
>
>
>
>
>
>
>
> On Sep 9, 2010, at 4:14 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>> >
>> > Here's a 48-tone version of the same thing.
>> >
>> > This combines an attempt at epimorphism with 48-tone fingering with 11 and 13-limit JI with a concession to the standard keyboard pattern of 12 notes.
>> >
>> > It's pretty learnable on a standard keyboard, because of the 48-note pattern and consistent 5ths.
>>
>> This should be easily learnable, and has a lot of nice 11-limit harmony:
>>
>> ! compton48.scl
>> !
>> Compton[48] 11-limit tweaked
>> 48
>> !
>> 50.02948
>> 66.62693
>> 83.40254
>> 100.00000
>> 150.02948
>> 166.62693
>> 183.40254
>> 200.00000
>> 250.02948
>> 266.62693
>> 283.40254
>> 300.00000
>> 350.02948
>> 366.62693
>> 383.40254
>> 400.00000
>> 450.02948
>> 466.62693
>> 483.40254
>> 500.00000
>> 550.02948
>> 566.62693
>> 583.40254
>> 600.00000
>> 650.02948
>> 666.62693
>> 683.40254
>> 700.00000
>> 750.02948
>> 766.62693
>> 783.40254
>> 800.00000
>> 850.02948
>> 866.62693
>> 883.40254
>> 900.00000
>> 950.02948
>> 966.62693
>> 983.40254
>> 1000.00000
>> 1050.02948
>> 1066.62693
>> 1083.40254
>> 1100.00000
>> 1150.02948
>> 1166.62693
>> 1183.40254
>> 1200.00000
>>
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗Andy <a_sparschuh@...>

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

Hi Caleb,
consider
http://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale

> It would be great to find one scale that is better than all the
> rest for a standard keyboard,
How about

/tuning/photos/album/515941611/pic/1703995626/view?picmode=&mode=tn&order=ordinal&start=1&count=20&dir=asc

http://www.synthtopia.com/content/2007/10/18/the-mysterious-terpstra-midi-keyboard/

or it's historically forerunner model the Bosanquet keyboard

http://sites.google.com/site/commator/flukes
" In the early 70's the 19 th century, this aggregate was tuned as a 53EDO system and each octave of its seven-row keyboard with many unison doubled tones counts in 84 keys. "

> allowing some modulation,
> giving good 7 and 11-limit harmony,
> with chains of 4ths and 5ths around 7-10 notes long on either
> side.

Why not empolying all the 53 times 4ths and 5ths?
alike in my recent rational 53-tone:

/tuning/topicId_90883.html#92003

> They don't have to be the same size, but they do have to finger the > same, and be acceptable, not wolves or voles.

Fully agreed!

!Sp53rat.scl
Sparschuh's [2010] rational 53-tone with some epimoric biased 5ths
53
!
65/64 ! A +01 D/
33/32 ! B +02 D+
26/25 ! C +03 EB
20/19 ! D +04 Eb
16/15 ! E +05 D#
13/12 ! F +06 D&
11/10 ! G +07 E+
10/9 ! H +08 E\
9/8 ! I +09 E.
8/7 ! J +10 E/
52/45 ! K +11 E+ = F-
7/6 ! L +12 F\
32/27 ! M +13 F.
6/5 ! N +14 F/
11/9 ! O +15 F+
16/13 ! P +16 GB
5/4 ! Q +17 Gb
81/64 ! R +18 F#
9/7 ! S +19 F&
57/44 ! T +20 G-
21/16 ! U +21 G\
4/3 ! V +22 G.
23/17 ! W +23 G/
11/8 ! X +24 G+
18/13 ! Y +25 AB
7/5 ! Z +26 Ab
10/7 ! z' -26 G#
13/9 ! y' -25 G&
16/11 ! x' -24 A-
34/23 ! w' -23 A\
3/2 ! v' -22 A.
32/21 ! u' -21 A/
88/57 ! t' -20 A+
14/9 ! s' -19 BB
128/81 ! r' -18 Bb
8/5 ! q' -17 A#
13/8 ! p' -16 A&
18/11 ! o' -15 B-
5/3 ! n' -14 B\
27/16 ! m' -13 B.
12/7 ! l' -12 B/
45/26 ! k' -11 B+ = C-
7/4 ! j' -10 C\
16/9 ! i' -09 C.
9/5 ! h' -08 C/
20/11 ! g' -07 C+
24/13 ! f' -06 DB
15/8 ! e' -05 Db
19/10 ! d' -04 C#
25/13 ! c' -03 C&
64/33 ! b' -02 D-
128/65 ! a' -01 D\
2/1 ! @' +-0 D.
!
!
![eof]

that satisfies and even exceeds all yours mentioned criterions.

bye
Andy

🔗caleb morgan <calebmrgn@...>

Hi Gene, I'll try what you've suggested.

The only thing special about 43 (or 42) is that I can fit 2 octaves into a standard Midi controller with 7 octaves plus 3 keys = 87 keys. I've got two of them.

Of course you're right that 43 doesn't work so well.

So, either it's back to something consistent with 41, or something that fingers inconsistently, or something with less that 2 octaves, or buying some new keyboards.

My latest, plan (see previous post) is to use one of your 46-note scales with 4 notes removed.

Alas, I insist on something very much like what Andy is proposing, but I want to throw out a few notes, or have my keyboard grow another octave.

Caught between horn and hardart.

Hey Andy, this looks good, but I didn't make it clear that I really want to fit two octaves--this has only become obvious to me recently.

On Sep 10, 2010, at 2:19 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > The "perfect" scale would be:
> >
> > Size: 43 notes
>
> That's not a particularly good staring point for scales with wide fifths, though stretching the octave could change that. You could just take 43 equal and stretch the octave to 1204 cents and see if you like that, I suppose. An alternative would be to take Meantone[43] and stretch it. Stick 81/80 and 126/125, or 81/80, 126/125 and 99/98 in Graham's magic box, and use those numbers. The octave stretch is less than a cent, which I personally think is a good thing. Putting in 126/125 and 176/175 leads to thrush temperament, which is considerably more accurate and does tend to wide major thirds, but not wide fifths or octaves.
>
> > octave: within 4 cents, preferably wide, say 1204.
>
> Four cents is a hell of a wide octave considering the tolerances you are talking about. I'm not sure what to do now, as I'm not really sure you want octaves so wide. I wasn't aware wide octaves was a consideration; I've been leaving them pure.
>
> > I think this is do-able. I think it's important, even.
>
> Why is it important? I don't see what is special about 43.
>
> > > ! compton48.scl
> > > !
> > > Compton[48] 11-limit tweaked
> > > 48
>
> What do you think of Compton[48]?
>
>

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗caleb morgan <calebmrgn@...>

Ok, once again I apologize for changing my mind.

If any reader is grumpy, just ignore this message, I'm obsessed and tired.

What are some variations on 43EDO that play with the octave size a little and have any reasonable-size 5ths, large or small?

I'm obsessed with getting 8:7 and 7:6 in tune, for some reason.

caleb

Oh, on another note, since 43 is a prime, any number mod 43 multiplied repeatedly by any N will give you all 43 numbers.

So, any member of something approximating 43EDO *could* be a generator.

for example, 12 steps in mod43 generates all 43 steps eventually.

k = 1 2 3 4 5 6 7 8 9 10
12*k= 12 24 36 5 17 29 41 10 22 34

And, I notice that changing the octave size (basically, the modulus, in a way) has a HUGE impact on the results.

Now, I find that I'm quite *unfussy* about the octave. So, it doesn't need to be that accurate.

Plus, I don't mind quite uneven step-sizes. It doesn't really have to be an EDO.

That's the wiggle room, if I could only grok it.

Some slightly off-EDO generator and some slightly off-1200 foldover might give me a good result.

Somehow, there's a generator/modulus combination that will give me 43 good notes.

This is a weird way to approach the problem, but it kind of makes sense for me.

I still don't understand how to use Graham's scripts, partly because I don't have Scala yet to plug the numbers into, and partly because I'm missing some theory about what the names mean and the results should be.

I suspect the understanding gap would be solved by getting Scala, much as LMSO has changed my thinking already.

2 3 5 7
[< 43 68 100 121 ]>

Tuning Map (cents)
<1199.495, 1896.875, 2789.522, 3375.322]

Is this telling me that my 5ths will be a little flat, along with the octave?

Why not have less accurate octaves and 5ths, and more accurate 7ths, somehow, and less even steps...??

Sorry for weird format.

caleb

On Sep 10, 2010, at 4:37 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> >
> > --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@> wrote:
> > >
> > > Hi Gene, I'll try what you've suggested.
> > >
> > > The only thing special about 43 (or 42) is that I can fit
> > > 2 octaves into a standard Midi controller with 7 octaves
> > > plus 3 keys = 87 keys. I've got two of them.
> >
> > Have you tried 41-ET? You'll get two octaves plus a few
> > spare notes, and completely consistent fingering.
>
> Studloco (Miracle[41]) is another interesting possibility; a slightly sharp octave would make sense.
>
>

🔗Mike Battaglia <battaglia01@...>

On Fri, Sep 10, 2010 at 7:57 PM, caleb morgan <calebmrgn@...> wrote:
>
> Ok, once again I apologize for changing my mind.
>
> If any reader is grumpy, just ignore this message, I'm obsessed and tired.

And how grumpy I am. I demand you stop talking about this tuning, so
that I can maintain the sanctity of My Inbox without having to learn
how to use the "filter" function that comes standard on every modern
email client.

Feel bad? You should feel bad. >:(

> What are some variations on 43EDO that play with the octave size a little and have any reasonable-size 5ths, large or small?
>
> I'm obsessed with getting 8:7 and 7:6 in tune, for some reason.
> caleb

What are your thoughts on Miracle[41], as Gene suggested? Or 41-equal,
as Carl suggested? Perhaps you could then create a circulating
temperament that's midway between those if you like the sound.

And then there's always 31-equal for almost perfect septimal harmonies
as well, with mostly passable 11-limit harmonies and not quite as good
13-limit harmonies. The fifth is a little bit flat as you no doubt
know. Perhaps some circulating 31-note temperament might be good to
think about as well.

Just out of curiosity, and I just went back through the email chain
and couldn't find an answer - what don't you like about 46-tet? Your
discussion about it kickstarted some playing around with it on my end
too... seems like not a bad tuning.

-Mike

🔗caleb morgan <calebmrgn@...>

(My favorite tunings--including many by the people who post here--O. Yarman, GWS, G. Secor, etc.

can be found here:

http://www.box.net/shared/m37jhti1og#/shared/m37jhti1og/1/50561828

Feel free to download or use any. Some are mine. Some are from the list. Some are from the Scala archives.)

No, I love 46EDO. But it has 3 too many notes to fit 2 octaves onto my Midi controller.
So, I could remove a few notes, but that would create the inconsistent-fingering problem.

For some reason, I'm fixated on using nearly my whole keyboard.

So, 41 would be a good choice.

I've got a lot of 41-tunings.

I was just hoping somehow that if I searched long enough, that I could do better somehow.

It's not even clear to me right now if this is just compulsion or if I'm after something rational and real.

I don't understand what Miracle 41 is. I'd love to know how to generate it, or at least get a Scala file.

I'm not clear at all on the Miracle concept.

I'd like to know how to use LMSO (or any glorified calculator), or the steps to using these scripts, such as the one below.

Say, for instance, this one:

8 & 19 & 43

Equal Temperament Mappings
2 3 5 7 11 13
[< 43 68 100 121 149 159 ]
< 19 30 44 53 65 70 ]
< 8 13 19 23 28 30 ]>

Reduced Mapping
2 3 5 7 11 13
[< 1 0 0 -1 -1 1 ]
< 0 1 1 1 0 1 ]
< 0 0 2 6 12 3 ]>

Generator Tunings (cents)
[1200.462, 1899.336, 445.662>

Step Tunings (cents)
[19.833, 16.261, 4.836>

Tuning Map (cents)
<1200.462, 1899.336, 2790.660, 3372.844, 4147.478, 4436.784]

Maybe I'm looking for a compromise between JI and 41 EDO or JI and 46 EDO,

The thing I don't like about 43 EDO is the 8:7 and 7:6 isn't very close.

Caleb

On Sep 10, 2010, at 8:31 PM, Mike Battaglia wrote:

> On Fri, Sep 10, 2010 at 7:57 PM, caleb morgan <calebmrgn@yahoo.com> wrote:
> >
> > Ok, once again I apologize for changing my mind.
> >
> > If any reader is grumpy, just ignore this message, I'm obsessed and tired.
>
> And how grumpy I am. I demand you stop talking about this tuning, so
> that I can maintain the sanctity of My Inbox without having to learn
> how to use the "filter" function that comes standard on every modern
> email client.
>
> Feel bad? You should feel bad. >:(
>
> > What are some variations on 43EDO that play with the octave size a little and have any reasonable-size 5ths, large or small?
> >
> > I'm obsessed with getting 8:7 and 7:6 in tune, for some reason.
> > caleb
>
> What are your thoughts on Miracle[41], as Gene suggested? Or 41-equal,
> as Carl suggested? Perhaps you could then create a circulating
> temperament that's midway between those if you like the sound.
>
> And then there's always 31-equal for almost perfect septimal harmonies
> as well, with mostly passable 11-limit harmonies and not quite as good
> 13-limit harmonies. The fifth is a little bit flat as you no doubt
> know. Perhaps some circulating 31-note temperament might be good to
> think about as well.
>
> Just out of curiosity, and I just went back through the email chain
> and couldn't find an answer - what don't you like about 46-tet? Your
> discussion about it kickstarted some playing around with it on my end
> too... seems like not a bad tuning.
>
> -Mike
>

🔗Mike Battaglia <battaglia01@...>

On Fri, Sep 10, 2010 at 8:47 PM, caleb morgan <calebmrgn@...> wrote:
>
> I don't understand what Miracle 41 is. I'd love to know how to generate it, or at least get a Scala file.
> I'm not clear at all on the Miracle concept.
> I'd like to know how to use LMSO (or any glorified calculator), or the steps to using these scripts, such as the one below.
> Say, for instance, this one:

Miracle is something I just started delving into recently, so you
might find the thread happening simultaneously about that to be
helpful. I am again explaining this both to make sure I have it right
and to try and recommunicate what I have learned.

Miracle is generated by an interval called the "secor." The secor is
defined as 1/6 of a perfect fifth - that is, it splits the fifth into
6 parts. 12-tet (or any chromatic scale of any meantone), on the other
hand, splits the fifth into 7 parts, which is a more familiar way of
thinking about the fifth being split up. This diverges with that
tradition.

As the secor makes its way through the fifth and beyond, it hits a
bunch of interesting intervals. The first is the secor itself, which
is right between 16/15 and 15/14 (and functions as both, hence
225/224, which is the difference between them, is tempered out). Two
secors makes a wide second, which turns out to be an almost perfect
8/7. Three is a neutral third, which turns out to be an almost perfect
11/9. Four is a wide fourth, which turns out to be an almost perfect
21/16. Five pretty much nails 7/5, and 6 gives you an almost perfect
3/2. Seven gives you 8/5, eight gives you 12/7, and 9 gives you 11/6.

Some common miracle tunings are 31 (which has its problems) and 41
(which also has its problems). Combine them and you get 72, which is
also a miracle, because of top secret witchcraft pagan voodoo. 72 is
more accurate than 31 or 41 for 11-limit harmony. The secor for
miracle in 72-tet is 7 steps of 72, which you can think of as one
12-tet half step with one little 72-tet step added (1/6 of a semitone)
added on top of it.

(Seriously, if you haven't played around with 72, you need to get
started. Get scala installed and fire away. I didn't understand
11-limit harmony at all until I started messing with it. A 12-tet
major third, minus 1 little 72-tet "step," is a "just" 5/4. A 12-tet
minor 7th, minus 2 little 72-tet "steps," is a "just" 7/4. A 12-tet
minor 7th plus one little 72-tet step is a "just" 9/5. A 12-tet
tritone minus 3 little 72-tet steps is a "just" 11/8. It's really,
really easy.)

Anyways, so the secor is 7/72. This generates a 10 note MOS with 1
large step at the end and 9 smaller steps leading into it, so
sssssssssL. Until recently, I didn't understand what the hell the
point of this scale was, since there's not a 4:5:6:7 tetrad in it. Go
to the next MOS, though (21 notes), and the picture becomes much more
clear:

! blackjack.scl
!
21 note MOS of "MIRACLE" temperament, Erlich & Keenan, miracle1.scl,TL 2-5-2001
21
! G
83.33333 ! G#v
116.66667 ! Ab^
200.00000 ! A
233.33333 ! A>
316.66667 ! Bb^
350.00000 ! B[
383.33333 ! Bv
466.66667 ! C<
500.00000 ! C
583.33333 ! C#v
616.66667 ! Db^
700.00000 ! D
733.33333 ! D>
816.66667 ! Eb^
850.00000 ! E[
933.33333 ! E>
966.66667 ! F<
1050.00000 ! F]
1083.33333 ! F#v
1166.66667 ! G<
2/1 ! G

This is the famous "blackjack" miracle scale, so named because it has
21 notes, and blackjack is if you have 21, and you get it. Anyway,
load this up in Scala (get Scala installed) and go to the Chromatic
Clavier and set the notation system to B72, which is Graham's decimal
system. You will quickly see what the point of this scale is, which is
that all of the intervals lying adjacent to this 10-note decimal set
are extremely useful. One "diatonic" step to the right of the neutral
third is now 5/4, one diatonic step left is 6/5, and all sorts of
useful intervals are all over the place. Near the narrow fourth is
4/3, and near the wide second is 9/8, and so on.

The next mos is 31 notes ("canasta"), which gives even more intervals,
and then you have 41 notes ("stud loco"). It should also ring a bell
that 31-tet and 41-tet both support miracle temperament. Either way,
Gene is suggesting you mess around with stud loco, and perhaps if it's
a bit too "rough" for you, you can average it with 41-equal and come
up with some circulating temperament or something like that.

Here's the stud loco one:

! miracle3.scl
!
41 out of 72-tET Pythagorean scale "Miracle/Studloco", Erlich/Keenan 2001
41
!
33.33330
66.66660
83.33337
116.66667
149.99997
183.33327
200.00004
233.33334
266.66664
299.99994
316.66671
350.00001
383.33331
416.66661
433.33338
466.66668
499.99998
533.33328
550.00005
583.33335
616.66665
649.99995
666.66672
700.00002
733.33332
766.66662
783.33339
816.66669
849.99999
883.33329
900.00006
933.33336
966.66666
999.99996
1016.66673
1050.00003
1083.33333
1116.66663
1133.33340
1166.66670
2/1

See how you like it. One caveat: there is apparently a better
generator for 13-limit stuff that's like a few fractions of a cent
away. I don't remember what it is. Hopefully someone here will
remember. I remember there was more than one comma that could be
tempered out to get to the 13-limit, and just did a search, and
couldn't find it. But this should get you started.

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗Herman Miller <hmiller@...>

caleb morgan wrote:
> I'm sorry for excessive posting if this has been the case.
> > I'm in a scale-design phase, which I hope to wrap up shortly.
> > It's been a process of learning what's possible, what's desirable, what's convenient, and what's easily learnable.
> > I've been circling 'round something. It involves JI/EDO hybrids of between 36 and 48 pitches.
> > 43 pitches turns out to fit two "octaves" perfectly, so it seems optimal for a number of reasons.
> > A standard keyboard is a given.
> > So, bear with me if I change my specs slightly one more time.
> > The "perfect" scale would be:
> > Size: 43 notes
> > Consistency: Not *entirely* consistent in smallest-step sizes, but fingering of closest equivalents to 3/2, 4/3, 9/8, 16/9, 5/4, 8/5, etc. would always be the *same* number of key-steps.
> > 5ths: wide but no wider than 704.35
> > 5/4's: wide by no more than 6 cents or dead on
> > 8/7, 7/6, 12/7, 7/4 off by no more than 6 cents
> > octave: within 4 cents, preferably wide, say 1204.
> > Consistency in every "key"*: *not* necessary. However, every "key" that has 1/1 as a member must be fairly in-tune, that is, have a good 3:4:5 triad.
> > There can be a number of "bad" keys, such as the one on the "second" degree of scale of 43.
> > There can be a very few "filler" notes for the sole purpose of making the fingering consistent. You might never use them.
> > The "error in the system" should be greatest around the keys that you wouldn't use as being harmonically-related to 1/1. So, /8, /9 etc to /11, perhaps even /13 would be accurate enough.
> > So would the "strange" tonalities on 9/8, and 5/4, and 27/16 and 5/3, even.
> > But some "keys" could um, suck, to put it bluntly.
> > So, it's possible to play this scale with good consonant 5:6:7:8:9:10:11 in at least 7 keys, with many more good but non OT scales available on other keys.
> > I think this is do-able. I think it's important, even.
> > *Some* beating in the 5's, 9's, 11's 13's, and 15's is even desirable, but not too much!
> > It may already exist.
> > It might look like a slight tempering of Partch's scale.
> > This is my current thinking.
>

Here's one that looks interesting.

3/2 is 701.96 cents.
5/4 is 4.78 cents sharp.
8/7, 7/6, 12/7, and 7/4 are all less than half a cent off.
The scale is strictly proper, distributional even, and Constant Structure.

Generators: 2/1, 391.095

! C:\music\scales\43-46.scl
!
43 notes of 43&46 regular temperament
43
!
26.71500
53.43000
80.14500
106.86000
133.57500
160.29000
187.00500
230.80500
257.52000
284.23500
310.95000
337.66500
364.38000
391.09500
417.81000
444.52500
471.24000
497.95500
524.67000
551.38500
578.10000
621.90000
648.61500
675.33000
702.04500
728.76000
755.47500
782.19000
808.90500
835.62000
862.33500
889.05000
915.76500
942.48000
969.19500
1012.99500
1039.71000
1066.42500
1093.14000
1119.85500
1146.57000
1173.28500
2/1

🔗Mike Battaglia <battaglia01@...>

🔗jonszanto <jszanto@...>

🔗Daniel Forró <dan.for@...>

🔗Mike Battaglia <battaglia01@...>

🔗Daniel Forró <dan.for@...>

🔗genewardsmith <genewardsmith@...>

🔗Daniel Forró <dan.for@...>

🔗caleb morgan <calebmrgn@...>

Outstanding Mike!

I think this should be included with the explanations on the Wiki and on the Xenharmony pages.

Really, really I do.

It makes sense to me in a way that more terse explanations don't.

It's going in my 'microtonal wisdom' file--which is just a file of copies of posts I found particularly helpful.

Thanks, I'll even play around with these scales. Some look like 36 EDO at first, but then I see some 76EDO in there.

Caleb

On Sep 10, 2010, at 9:34 PM, Mike Battaglia wrote:

> On Fri, Sep 10, 2010 at 8:47 PM, caleb morgan <calebmrgn@...> wrote:
> >
> > I don't understand what Miracle 41 is. I'd love to know how to generate it, or at least get a Scala file.
> > I'm not clear at all on the Miracle concept.
> > I'd like to know how to use LMSO (or any glorified calculator), or the steps to using these scripts, such as the one below.
> > Say, for instance, this one:
>
> Miracle is something I just started delving into recently, so you
> might find the thread happening simultaneously about that to be
> helpful. I am again explaining this both to make sure I have it right
> and to try and recommunicate what I have learned.
>
> Miracle is generated by an interval called the "secor." The secor is
> defined as 1/6 of a perfect fifth - that is, it splits the fifth into
> 6 parts. 12-tet (or any chromatic scale of any meantone), on the other
> hand, splits the fifth into 7 parts, which is a more familiar way of
> thinking about the fifth being split up. This diverges with that
> tradition.
>
> As the secor makes its way through the fifth and beyond, it hits a
> bunch of interesting intervals. The first is the secor itself, which
> is right between 16/15 and 15/14 (and functions as both, hence
> 225/224, which is the difference between them, is tempered out). Two
> secors makes a wide second, which turns out to be an almost perfect
> 8/7. Three is a neutral third, which turns out to be an almost perfect
> 11/9. Four is a wide fourth, which turns out to be an almost perfect
> 21/16. Five pretty much nails 7/5, and 6 gives you an almost perfect
> 3/2. Seven gives you 8/5, eight gives you 12/7, and 9 gives you 11/6.
>
> Some common miracle tunings are 31 (which has its problems) and 41
> (which also has its problems). Combine them and you get 72, which is
> also a miracle, because of top secret witchcraft pagan voodoo. 72 is
> more accurate than 31 or 41 for 11-limit harmony. The secor for
> miracle in 72-tet is 7 steps of 72, which you can think of as one
> 12-tet half step with one little 72-tet step added (1/6 of a semitone)
> added on top of it.
>
> (Seriously, if you haven't played around with 72, you need to get
> started. Get scala installed and fire away. I didn't understand
> 11-limit harmony at all until I started messing with it. A 12-tet
> major third, minus 1 little 72-tet "step," is a "just" 5/4. A 12-tet
> minor 7th, minus 2 little 72-tet "steps," is a "just" 7/4. A 12-tet
> minor 7th plus one little 72-tet step is a "just" 9/5. A 12-tet
> tritone minus 3 little 72-tet steps is a "just" 11/8. It's really,
> really easy.)
>
> Anyways, so the secor is 7/72. This generates a 10 note MOS with 1
> large step at the end and 9 smaller steps leading into it, so
> sssssssssL. Until recently, I didn't understand what the hell the
> point of this scale was, since there's not a 4:5:6:7 tetrad in it. Go
> to the next MOS, though (21 notes), and the picture becomes much more
> clear:
>
> ! blackjack.scl
> !
> 21 note MOS of "MIRACLE" temperament, Erlich & Keenan, miracle1.scl,TL 2-5-2001
> 21
> ! G
> 83.33333 ! G#v
> 116.66667 ! Ab^
> 200.00000 ! A
> 233.33333 ! A>
> 316.66667 ! Bb^
> 350.00000 ! B[
> 383.33333 ! Bv
> 466.66667 ! C<
> 500.00000 ! C
> 583.33333 ! C#v
> 616.66667 ! Db^
> 700.00000 ! D
> 733.33333 ! D>
> 816.66667 ! Eb^
> 850.00000 ! E[
> 933.33333 ! E>
> 966.66667 ! F<
> 1050.00000 ! F]
> 1083.33333 ! F#v
> 1166.66667 ! G<
> 2/1 ! G
>
> This is the famous "blackjack" miracle scale, so named because it has
> 21 notes, and blackjack is if you have 21, and you get it. Anyway,
> load this up in Scala (get Scala installed) and go to the Chromatic
> Clavier and set the notation system to B72, which is Graham's decimal
> system. You will quickly see what the point of this scale is, which is
> that all of the intervals lying adjacent to this 10-note decimal set
> are extremely useful. One "diatonic" step to the right of the neutral
> third is now 5/4, one diatonic step left is 6/5, and all sorts of
> useful intervals are all over the place. Near the narrow fourth is
> 4/3, and near the wide second is 9/8, and so on.
>
> The next mos is 31 notes ("canasta"), which gives even more intervals,
> and then you have 41 notes ("stud loco"). It should also ring a bell
> that 31-tet and 41-tet both support miracle temperament. Either way,
> Gene is suggesting you mess around with stud loco, and perhaps if it's
> a bit too "rough" for you, you can average it with 41-equal and come
> up with some circulating temperament or something like that.
>
> Here's the stud loco one:
>
> ! miracle3.scl
> !
> 41 out of 72-tET Pythagorean scale "Miracle/Studloco", Erlich/Keenan 2001
> 41
> !
> 33.33330
> 66.66660
> 83.33337
> 116.66667
> 149.99997
> 183.33327
> 200.00004
> 233.33334
> 266.66664
> 299.99994
> 316.66671
> 350.00001
> 383.33331
> 416.66661
> 433.33338
> 466.66668
> 499.99998
> 533.33328
> 550.00005
> 583.33335
> 616.66665
> 649.99995
> 666.66672
> 700.00002
> 733.33332
> 766.66662
> 783.33339
> 816.66669
> 849.99999
> 883.33329
> 900.00006
> 933.33336
> 966.66666
> 999.99996
> 1016.66673
> 1050.00003
> 1083.33333
> 1116.66663
> 1133.33340
> 1166.66670
> 2/1
>
> See how you like it. One caveat: there is apparently a better
> generator for 13-limit stuff that's like a few fractions of a cent
> away. I don't remember what it is. Hopefully someone here will
> remember. I remember there was more than one comma that could be
> tempered out to get to the 13-limit, and just did a search, and
> couldn't find it. But this should get you started.
>
> -Mike
>

🔗caleb morgan <calebmrgn@...>

Damn, Herman, that's good! That's what I was looking for when I posted this. I might have found it myself, (Caleb says this with slight disappointment but real appreciation.)

I wish I knew the thought process that lead you there.

Maybe I will, soon.

Thanks a lot, that's going in the favorite scales folder.

Caleb

On Sep 10, 2010, at 10:54 PM, Herman Miller wrote:

> caleb morgan wrote:
>> I'm sorry for excessive posting if this has been the case.
>>
>> I'm in a scale-design phase, which I hope to wrap up shortly.
>>
>> It's been a process of learning what's possible, what's desirable, what's convenient, and what's easily learnable.
>>
>> I've been circling 'round something. It involves JI/EDO hybrids of between 36 and 48 pitches.
>>
>> 43 pitches turns out to fit two "octaves" perfectly, so it seems optimal for a number of reasons.
>>
>> A standard keyboard is a given.
>>
>> So, bear with me if I change my specs slightly one more time.
>>
>> The "perfect" scale would be:
>>
>> Size: 43 notes
>>
>> Consistency: Not *entirely* consistent in smallest-step sizes, but fingering of closest equivalents to 3/2, 4/3, 9/8, 16/9, 5/4, 8/5, etc. would always be the *same* number of key-steps.
>>
>> 5ths: wide but no wider than 704.35
>>
>> 5/4's: wide by no more than 6 cents or dead on
>>
>> 8/7, 7/6, 12/7, 7/4 off by no more than 6 cents
>>
>> octave: within 4 cents, preferably wide, say 1204.
>>
>> Consistency in every "key"*: *not* necessary. However, every "key" that has 1/1 as a member must be fairly in-tune, that is, have a good 3:4:5 triad.
>>
>> There can be a number of "bad" keys, such as the one on the "second" degree of scale of 43.
>>
>> There can be a very few "filler" notes for the sole purpose of making the fingering consistent. You might never use them.
>>
>> The "error in the system" should be greatest around the keys that you wouldn't use as being harmonically-related to 1/1. So, /8, /9 etc to /11, perhaps even /13 would be accurate enough.
>>
>> So would the "strange" tonalities on 9/8, and 5/4, and 27/16 and 5/3, even.
>>
>> But some "keys" could um, suck, to put it bluntly.
>>
>> So, it's possible to play this scale with good consonant 5:6:7:8:9:10:11 in at least 7 keys, with many more good but non OT scales available on other keys.
>>
>> I think this is do-able. I think it's important, even.
>>
>> *Some* beating in the 5's, 9's, 11's 13's, and 15's is even desirable, but not too much!
>>
>> It may already exist.
>>
>> It might look like a slight tempering of Partch's scale.
>>
>> This is my current thinking.
>>
>
> Here's one that looks interesting.
>
> 3/2 is 701.96 cents.
> 5/4 is 4.78 cents sharp.
> 8/7, 7/6, 12/7, and 7/4 are all less than half a cent off.
> The scale is strictly proper, distributional even, and Constant Structure.
>
> Generators: 2/1, 391.095
>
> ! C:\music\scales\43-46.scl
> !
> 43 notes of 43&46 regular temperament
> 43
> !
> 26.71500
> 53.43000
> 80.14500
> 106.86000
> 133.57500
> 160.29000
> 187.00500
> 230.80500
> 257.52000
> 284.23500
> 310.95000
> 337.66500
> 364.38000
> 391.09500
> 417.81000
> 444.52500
> 471.24000
> 497.95500
> 524.67000
> 551.38500
> 578.10000
> 621.90000
> 648.61500
> 675.33000
> 702.04500
> 728.76000
> 755.47500
> 782.19000
> 808.90500
> 835.62000
> 862.33500
> 889.05000
> 915.76500
> 942.48000
> 969.19500
> 1012.99500
> 1039.71000
> 1066.42500
> 1093.14000
> 1119.85500
> 1146.57000
> 1173.28500
> 2/1
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>

🔗caleb morgan <calebmrgn@...>

Caleb plays scale.

Oops, 230.805 is doing double duty as 9/8 and 8/7!

Jon Szanto was right: Compromise is for domestic felines.

Not *this* feral cat.

Caleb

On Sep 11, 2010, at 4:59 AM, caleb morgan wrote:

> Damn, Herman, that's good! That's what I was looking for when I posted this. I might have found it myself, (Caleb says this with slight disappointment but real appreciation.)
>
> I wish I knew the thought process that lead you there.
>
> Maybe I will, soon.
>
> Thanks a lot, that's going in the favorite scales folder.
>
> Caleb
>
> On Sep 10, 2010, at 10:54 PM, Herman Miller wrote:
>
> > caleb morgan wrote:
> >> I'm sorry for excessive posting if this has been the case.
> >>
> >> I'm in a scale-design phase, which I hope to wrap up shortly.
> >>
> >> It's been a process of learning what's possible, what's desirable, what's convenient, and what's easily learnable.
> >>
> >> I've been circling 'round something. It involves JI/EDO hybrids of between 36 and 48 pitches.
> >>
> >> 43 pitches turns out to fit two "octaves" perfectly, so it seems optimal for a number of reasons.
> >>
> >> A standard keyboard is a given.
> >>
> >> So, bear with me if I change my specs slightly one more time.
> >>
> >> The "perfect" scale would be:
> >>
> >> Size: 43 notes
> >>
> >> Consistency: Not *entirely* consistent in smallest-step sizes, but fingering of closest equivalents to 3/2, 4/3, 9/8, 16/9, 5/4, 8/5, etc. would always be the *same* number of key-steps.
> >>
> >> 5ths: wide but no wider than 704.35
> >>
> >> 5/4's: wide by no more than 6 cents or dead on
> >>
> >> 8/7, 7/6, 12/7, 7/4 off by no more than 6 cents
> >>
> >> octave: within 4 cents, preferably wide, say 1204.
> >>
> >> Consistency in every "key"*: *not* necessary. However, every "key" that has 1/1 as a member must be fairly in-tune, that is, have a good 3:4:5 triad.
> >>
> >> There can be a number of "bad" keys, such as the one on the "second" degree of scale of 43.
> >>
> >> There can be a very few "filler" notes for the sole purpose of making the fingering consistent. You might never use them.
> >>
> >> The "error in the system" should be greatest around the keys that you wouldn't use as being harmonically-related to 1/1. So, /8, /9 etc to /11, perhaps even /13 would be accurate enough.
> >>
> >> So would the "strange" tonalities on 9/8, and 5/4, and 27/16 and 5/3, even.
> >>
> >> But some "keys" could um, suck, to put it bluntly.
> >>
> >> So, it's possible to play this scale with good consonant 5:6:7:8:9:10:11 in at least 7 keys, with many more good but non OT scales available on other keys.
> >>
> >> I think this is do-able. I think it's important, even.
> >>
> >> *Some* beating in the 5's, 9's, 11's 13's, and 15's is even desirable, but not too much!
> >>
> >> It may already exist.
> >>
> >> It might look like a slight tempering of Partch's scale.
> >>
> >> This is my current thinking.
> >>
> >
> > Here's one that looks interesting.
> >
> > 3/2 is 701.96 cents.
> > 5/4 is 4.78 cents sharp.
> > 8/7, 7/6, 12/7, and 7/4 are all less than half a cent off.
> > The scale is strictly proper, distributional even, and Constant Structure.
> >
> > Generators: 2/1, 391.095
> >
> > ! C:\music\scales\43-46.scl
> > !
> > 43 notes of 43&46 regular temperament
> > 43
> > !
> > 26.71500
> > 53.43000
> > 80.14500
> > 106.86000
> > 133.57500
> > 160.29000
> > 187.00500
> > 230.80500
> > 257.52000
> > 284.23500
> > 310.95000
> > 337.66500
> > 364.38000
> > 391.09500
> > 417.81000
> > 444.52500
> > 471.24000
> > 497.95500
> > 524.67000
> > 551.38500
> > 578.10000
> > 621.90000
> > 648.61500
> > 675.33000
> > 702.04500
> > 728.76000
> > 755.47500
> > 782.19000
> > 808.90500
> > 835.62000
> > 862.33500
> > 889.05000
> > 915.76500
> > 942.48000
> > 969.19500
> > 1012.99500
> > 1039.71000
> > 1066.42500
> > 1093.14000
> > 1119.85500
> > 1146.57000
> > 1173.28500
> > 2/1
> >
> >
> >
> >
> > ------------------------------------
> >
> > You can configure your subscription by sending an empty email to one
> > of these addresses (from the address at which you receive the list):
> > tuning-subscribe@yahoogroups.com - join the tuning group.
> > tuning-unsubscribe@yahoogroups.com - leave the group.
> > tuning-nomail@yahoogroups.com - turn off mail from the group.
> > tuning-digest@yahoogroups.com - set group to send daily digests.
> > tuning-normal@yahoogroups.com - set group to send individual emails.
> > tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> >
> >
> >
>
>

🔗caleb morgan <calebmrgn@...>

Awright, building on the idea of a 13-limit tonality diamond in 72 EDO:

1/1 0. 200. 383.333 550 700 833.333 966.667 1083.333

8:9 1000. 0., 183.333 350 500 633.333 766.667 883.333

4:5 816.667 1016.667 0 166.667 316.667 450. 583.333 700

8:11 650. 850. 1033.333 0 150. 283.333 416.667 533.333

4:3 500. 700. 883.333 1050 0 133.333 266.667 383.333

16:13 366.667 566.667, 750. 916.667 1066.667 0 133.333 250.

8:7 233.333, 433.333 616.667 783.333 933.333 1066.667 0 116.667

8:15 116.667 316.667 500. 666.667 816.667 950. 1083.333, 0

arranged as a scale it gives us 45:

!0.
116.667
133.333
150.0
166.667
183.333
!
200.0
233.333
250.0
266.667
283.333
316.667
!
350.0
366.667
383.333
416.667
433.333
450.0
!
500.0
533.333
550.0
566.667
583.333
616.667
!
633.333
650.0
666.667
700.
750.
766.667
!
783.333
816.667
833.333
850.0
883.333
916.667
!
933.333
950.0
966.667
1000.
1016.667
1033.333
!
1050.0
1066.667
1083.333
!
1200

from which I snip 2 tones to give 43:

!43Government Work
43-pitch near-complete 13-limit in 72EDO
43
!0.
116.667
133.333
150.0
166.667
183.333
!
200.0
233.333
250.0
266.667
283.333
316.667
!
350.0
366.667
383.333
416.667
433.333
450.0
!
500.0
550.0
566.667
583.333
616.667
!
633.333
650.0
666.667
700.
750.
766.667
!
783.333
816.667
833.333
850.0
883.333
916.667
!
933.333
966.667
1000.
1016.667
1033.333
!
1050.0
1066.667
1083.333
!
1200

On Sep 11, 2010, at 5:36 AM, caleb morgan wrote:

> Caleb plays scale.
>
>
> Oops, 230.805 is doing double duty as 9/8 and 8/7!
>
> Jon Szanto was right: Compromise is for domestic felines.
>
> Not *this* feral cat.
>
> Caleb
>
>
>
> On Sep 11, 2010, at 4:59 AM, caleb morgan wrote:
>
>>
>> Damn, Herman, that's good! That's what I was looking for when I posted this. I might have found it myself, (Caleb says this with slight disappointment but real appreciation.)
>>
>> I wish I knew the thought process that lead you there.
>>
>> Maybe I will, soon.
>>
>> Thanks a lot, that's going in the favorite scales folder.
>>
>> Caleb
>>
>> On Sep 10, 2010, at 10:54 PM, Herman Miller wrote:
>>
>> > caleb morgan wrote:
>> >> I'm sorry for excessive posting if this has been the case.
>> >>
>> >> I'm in a scale-design phase, which I hope to wrap up shortly.
>> >>
>> >> It's been a process of learning what's possible, what's desirable, what's convenient, and what's easily learnable.
>> >>
>> >> I've been circling 'round something. It involves JI/EDO hybrids of between 36 and 48 pitches.
>> >>
>> >> 43 pitches turns out to fit two "octaves" perfectly, so it seems optimal for a number of reasons.
>> >>
>> >> A standard keyboard is a given.
>> >>
>> >> So, bear with me if I change my specs slightly one more time.
>> >>
>> >> The "perfect" scale would be:
>> >>
>> >> Size: 43 notes
>> >>
>> >> Consistency: Not *entirely* consistent in smallest-step sizes, but fingering of closest equivalents to 3/2, 4/3, 9/8, 16/9, 5/4, 8/5, etc. would always be the *same* number of key-steps.
>> >>
>> >> 5ths: wide but no wider than 704.35
>> >>
>> >> 5/4's: wide by no more than 6 cents or dead on
>> >>
>> >> 8/7, 7/6, 12/7, 7/4 off by no more than 6 cents
>> >>
>> >> octave: within 4 cents, preferably wide, say 1204.
>> >>
>> >> Consistency in every "key"*: *not* necessary. However, every "key" that has 1/1 as a member must be fairly in-tune, that is, have a good 3:4:5 triad.
>> >>
>> >> There can be a number of "bad" keys, such as the one on the "second" degree of scale of 43.
>> >>
>> >> There can be a very few "filler" notes for the sole purpose of making the fingering consistent. You might never use them.
>> >>
>> >> The "error in the system" should be greatest around the keys that you wouldn't use as being harmonically-related to 1/1. So, /8, /9 etc to /11, perhaps even /13 would be accurate enough.
>> >>
>> >> So would the "strange" tonalities on 9/8, and 5/4, and 27/16 and 5/3, even.
>> >>
>> >> But some "keys" could um, suck, to put it bluntly.
>> >>
>> >> So, it's possible to play this scale with good consonant 5:6:7:8:9:10:11 in at least 7 keys, with many more good but non OT scales available on other keys.
>> >>
>> >> I think this is do-able. I think it's important, even.
>> >>
>> >> *Some* beating in the 5's, 9's, 11's 13's, and 15's is even desirable, but not too much!
>> >>
>> >> It may already exist.
>> >>
>> >> It might look like a slight tempering of Partch's scale.
>> >>
>> >> This is my current thinking.
>> >>
>> >
>> > Here's one that looks interesting.
>> >
>> > 3/2 is 701.96 cents.
>> > 5/4 is 4.78 cents sharp.
>> > 8/7, 7/6, 12/7, and 7/4 are all less than half a cent off.
>> > The scale is strictly proper, distributional even, and Constant Structure.
>> >
>> > Generators: 2/1, 391.095
>> >
>> > ! C:\music\scales\43-46.scl
>> > !
>> > 43 notes of 43&46 regular temperament
>> > 43
>> > !
>> > 26.71500
>> > 53.43000
>> > 80.14500
>> > 106.86000
>> > 133.57500
>> > 160.29000
>> > 187.00500
>> > 230.80500
>> > 257.52000
>> > 284.23500
>> > 310.95000
>> > 337.66500
>> > 364.38000
>> > 391.09500
>> > 417.81000
>> > 444.52500
>> > 471.24000
>> > 497.95500
>> > 524.67000
>> > 551.38500
>> > 578.10000
>> > 621.90000
>> > 648.61500
>> > 675.33000
>> > 702.04500
>> > 728.76000
>> > 755.47500
>> > 782.19000
>> > 808.90500
>> > 835.62000
>> > 862.33500
>> > 889.05000
>> > 915.76500
>> > 942.48000
>> > 969.19500
>> > 1012.99500
>> > 1039.71000
>> > 1066.42500
>> > 1093.14000
>> > 1119.85500
>> > 1146.57000
>> > 1173.28500
>> > 2/1
>> >
>> >
>> >
>> >
>> > ------------------------------------
>> >
>> > You can configure your subscription by sending an empty email to one
>> > of these addresses (from the address at which you receive the list):
>> > tuning-subscribe@yahoogroups.com - join the tuning group.
>> > tuning-unsubscribe@yahoogroups.com - leave the group.
>> > tuning-nomail@yahoogroups.com - turn off mail from the group.
>> > tuning-digest@yahoogroups.com - set group to send daily digests.
>> > tuning-normal@yahoogroups.com - set group to send individual emails.
>> > tuning-help@yahoogroups.com - receive general help information.
>> > Yahoo! Groups Links
>> >
>> >
>> >
>>
>
>
>

🔗Michael <djtrancendance@...>

🔗caleb morgan <calebmrgn@...>

as if to prove GWS right yet again, I'm posting, for your edification, a new kind of scale,

the MOI? (Moment of Incompetence) scale.

!43-tone Moment of Incompetence (MOI) scale
MOI?
43
!0.
28.
43.7
70.
92.5
165.
182.4
203.
232.2
266.9
295.5
308.
315.64
386.3
406.
417.5
454.2
498.5
499.
536.9
551.3
591.
!
609.
648.7
663.
701.5
745.7
772.
794.
813.686
867.
884.
904.5
935.1
969.8
997.
1008.
1049.4
1088.27
1107.5
1119.
1155.
1176.
1202.

On Sep 11, 2010, at 7:29 AM, caleb morgan wrote:

>
>
> Awright, building on the idea of a 13-limit tonality diamond in 72 EDO:
>
> 1/1 0. 200. 383.333 550 700 833.333 966.667 1083.333
>
> 8:9 1000. 0., 183.333 350 500 633.333 766.667 883.333
>
> 4:5 816.667 1016.667 0 166.667 316.667 450. 583.333 700
>
> 8:11 650. 850. 1033.333 0 150. 283.333 416.667 533.333
>
> 4:3 500. 700. 883.333 1050 0 133.333 266.667 383.333
>
> 16:13 366.667 566.667, 750. 916.667 1066.667 0 133.333 250.
>
> 8:7 233.333, 433.333 616.667 783.333 933.333 1066.667 0 116.667
>
> 8:15 116.667 316.667 500. 666.667 816.667 950. 1083.333, 0
>
>
> arranged as a scale it gives us 45:
>
>
> !0.
> 116.667
> 133.333
> 150.0
> 166.667
> 183.333
> !
> 200.0
> 233.333
> 250.0
> 266.667
> 283.333
> 316.667
> !
> 350.0
> 366.667
> 383.333
> 416.667
> 433.333
> 450.0
> !
> 500.0
> 533.333
> 550.0
> 566.667
> 583.333
> 616.667
> !
> 633.333
> 650.0
> 666.667
> 700.
> 750.
> 766.667
> !
> 783.333
> 816.667
> 833.333
> 850.0
> 883.333
> 916.667
> !
> 933.333
> 950.0
> 966.667
> 1000.
> 1016.667
> 1033.333
> !
> 1050.0
> 1066.667
> 1083.333
> !
> 1200
>
>
> from which I snip 2 tones to give 43:
>
> !43Government Work
> 43-pitch near-complete 13-limit in 72EDO
> 43
> !0.
> 116.667
> 133.333
> 150.0
> 166.667
> 183.333
> !
> 200.0
> 233.333
> 250.0
> 266.667
> 283.333
> 316.667
> !
> 350.0
> 366.667
> 383.333
> 416.667
> 433.333
> 450.0
> !
> 500.0
> 550.0
> 566.667
> 583.333
> 616.667
> !
> 633.333
> 650.0
> 666.667
> 700.
> 750.
> 766.667
> !
> 783.333
> 816.667
> 833.333
> 850.0
> 883.333
> 916.667
> !
> 933.333
> 966.667
> 1000.
> 1016.667
> 1033.333
> !
> 1050.0
> 1066.667
> 1083.333
> !
> 1200
>
>
>
>
>
> On Sep 11, 2010, at 5:36 AM, caleb morgan wrote:
>
>>
>> Caleb plays scale.
>>
>>
>> Oops, 230.805 is doing double duty as 9/8 and 8/7!
>>
>> Jon Szanto was right: Compromise is for domestic felines.
>>
>> Not *this* feral cat.
>>
>> Caleb
>>
>>
>>
>> On Sep 11, 2010, at 4:59 AM, caleb morgan wrote:
>>
>>>
>>> Damn, Herman, that's good! That's what I was looking for when I posted this. I might have found it myself, (Caleb says this with slight disappointment but real appreciation.)
>>>
>>> I wish I knew the thought process that lead you there.
>>>
>>> Maybe I will, soon.
>>>
>>> Thanks a lot, that's going in the favorite scales folder.
>>>
>>> Caleb
>>>
>>> On Sep 10, 2010, at 10:54 PM, Herman Miller wrote:
>>>
>>> > caleb morgan wrote:
>>> >> I'm sorry for excessive posting if this has been the case.
>>> >>
>>> >> I'm in a scale-design phase, which I hope to wrap up shortly.
>>> >>
>>> >> It's been a process of learning what's possible, what's desirable, what's convenient, and what's easily learnable.
>>> >>
>>> >> I've been circling 'round something. It involves JI/EDO hybrids of between 36 and 48 pitches.
>>> >>
>>> >> 43 pitches turns out to fit two "octaves" perfectly, so it seems optimal for a number of reasons.
>>> >>
>>> >> A standard keyboard is a given.
>>> >>
>>> >> So, bear with me if I change my specs slightly one more time.
>>> >>
>>> >> The "perfect" scale would be:
>>> >>
>>> >> Size: 43 notes
>>> >>
>>> >> Consistency: Not *entirely* consistent in smallest-step sizes, but fingering of closest equivalents to 3/2, 4/3, 9/8, 16/9, 5/4, 8/5, etc. would always be the *same* number of key-steps.
>>> >>
>>> >> 5ths: wide but no wider than 704.35
>>> >>
>>> >> 5/4's: wide by no more than 6 cents or dead on
>>> >>
>>> >> 8/7, 7/6, 12/7, 7/4 off by no more than 6 cents
>>> >>
>>> >> octave: within 4 cents, preferably wide, say 1204.
>>> >>
>>> >> Consistency in every "key"*: *not* necessary. However, every "key" that has 1/1 as a member must be fairly in-tune, that is, have a good 3:4:5 triad.
>>> >>
>>> >> There can be a number of "bad" keys, such as the one on the "second" degree of scale of 43.
>>> >>
>>> >> There can be a very few "filler" notes for the sole purpose of making the fingering consistent. You might never use them.
>>> >>
>>> >> The "error in the system" should be greatest around the keys that you wouldn't use as being harmonically-related to 1/1. So, /8, /9 etc to /11, perhaps even /13 would be accurate enough.
>>> >>
>>> >> So would the "strange" tonalities on 9/8, and 5/4, and 27/16 and 5/3, even.
>>> >>
>>> >> But some "keys" could um, suck, to put it bluntly.
>>> >>
>>> >> So, it's possible to play this scale with good consonant 5:6:7:8:9:10:11 in at least 7 keys, with many more good but non OT scales available on other keys.
>>> >>
>>> >> I think this is do-able. I think it's important, even.
>>> >>
>>> >> *Some* beating in the 5's, 9's, 11's 13's, and 15's is even desirable, but not too much!
>>> >>
>>> >> It may already exist.
>>> >>
>>> >> It might look like a slight tempering of Partch's scale.
>>> >>
>>> >> This is my current thinking.
>>> >>
>>> >
>>> > Here's one that looks interesting.
>>> >
>>> > 3/2 is 701.96 cents.
>>> > 5/4 is 4.78 cents sharp.
>>> > 8/7, 7/6, 12/7, and 7/4 are all less than half a cent off.
>>> > The scale is strictly proper, distributional even, and Constant Structure.
>>> >
>>> > Generators: 2/1, 391.095
>>> >
>>> > ! C:\music\scales\43-46.scl
>>> > !
>>> > 43 notes of 43&46 regular temperament
>>> > 43
>>> > !
>>> > 26.71500
>>> > 53.43000
>>> > 80.14500
>>> > 106.86000
>>> > 133.57500
>>> > 160.29000
>>> > 187.00500
>>> > 230.80500
>>> > 257.52000
>>> > 284.23500
>>> > 310.95000
>>> > 337.66500
>>> > 364.38000
>>> > 391.09500
>>> > 417.81000
>>> > 444.52500
>>> > 471.24000
>>> > 497.95500
>>> > 524.67000
>>> > 551.38500
>>> > 578.10000
>>> > 621.90000
>>> > 648.61500
>>> > 675.33000
>>> > 702.04500
>>> > 728.76000
>>> > 755.47500
>>> > 782.19000
>>> > 808.90500
>>> > 835.62000
>>> > 862.33500
>>> > 889.05000
>>> > 915.76500
>>> > 942.48000
>>> > 969.19500
>>> > 1012.99500
>>> > 1039.71000
>>> > 1066.42500
>>> > 1093.14000
>>> > 1119.85500
>>> > 1146.57000
>>> > 1173.28500
>>> > 2/1
>>> >
>>> >
>>> >
>>> >
>>> > ------------------------------------
>>> >
>>> > You can configure your subscription by sending an empty email to one
>>> > of these addresses (from the address at which you receive the list):
>>> > tuning-subscribe@yahoogroups.com - join the tuning group.
>>> > tuning-unsubscribe@yahoogroups.com - leave the group.
>>> > tuning-nomail@yahoogroups.com - turn off mail from the group.
>>> > tuning-digest@yahoogroups.com - set group to send daily digests.
>>> > tuning-normal@yahoogroups.com - set group to send individual emails.
>>> > tuning-help@yahoogroups.com - receive general help information.
>>> > Yahoo! Groups Links
>>> >
>>> >
>>> >
>>>
>>
>>
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗caleb morgan <calebmrgn@...>

Here's something somewhere between serious and not serious.

The cent-values of 44EDO were generated, it was observed that most the odd-numbered pitches from 0 cents (27.27, 81.818, etc.) would generate 44EDO. Then values slightly off from those were tried. 190.909 was adjusted to be 190.3, then this was multiplied by n's up to 44.

These values were repeated at the octave, removing the next-to last pitch to give a 4:1 on the highest of 88 keys.

It *does* hit some good pitches and have some regularity.

Otherwise, it's kind of wacka-diddly.

So, in the spirit of "Flanders", I give you:

HiDiddlyHonegger.

!190.3 Gen 44 pitches, with top oct
HiDiddlyHonegger
88
!0.,
15.7,
58.2,
73.9,
116.4,
132.1,
174.6,
190.3,
206.,
248.5,
264.2,
306.7,
322.4,
364.9,
380.6,
396.3,
438.8,
454.5,
497.,
512.7,
555.2,
570.9,
586.6,
629.1,
644.8,
687.3,
703.,
745.5,
761.2,
803.7,
819.4,
835.1,
877.6,
893.3,
935.8,
951.5,
994.,
1009.7,
1025.4,
1067.9,
1083.6,
1126.1,
1141.8,
1184.3
1200
1215.7,
1258.2,
1273.9,
1316.4,
1332.1,
1374.6,
1390.3,
1406.,
1448.5,
1464.2,
1506.7,
1522.4,
1564.9,
1580.6,
1596.3,
1638.8,
1654.5,
1697.,
1712.7,
1755.2,
1770.9,
1786.6,
1829.1,
1844.8,
1887.3,
1903.,
1945.5,
1961.2,
2003.7,
2019.4,
2035.1,
2077.6,
2093.3,
2135.8,
2151.5,
2194.,
2209.7,
2225.4,
2267.9,
2283.6,
2326.1,
2341.8,
2400
2415.7

On Sep 11, 2010, at 2:02 PM, Mike Battaglia wrote:

> On Sat, Sep 11, 2010 at 1:48 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
> >
> > > !43-tone Moment of Incompetence (MOI) scale
> >
> > > 498.5
> > > 499.
> >
> > Is this the actual Moment of Incompetence?
>
> Hahaha! And then there were 42...
>
> Although if that were something like 478.5 it would fit the pattern better.
>
> -Mike
>

🔗caleb morgan <calebmrgn@...>

I've gone back and forth a lot, and hopefully I've learned something.

Tuning takes a mix of factors into account, from the mathematical to the contingent.

41 pitches per 2/1 seems to be the best framework for my future work.

With a consistent keyboard pattern, I don't have to learn a dozen new "languages"--I only have to learn one.

41 fits two "octaves" into an 88-note keyboard. I have 2 keyboards for 4 octaves--about the range of an electric guitar.

Any member of 41 can serve as a generator for any of the others, since 41 is prime.

With a program like LMSO or Scala, it's easy to generate variations on 41 quickly.

Each generator--if the value is changed slightly--can produce different patterns of slight distortion or perturbation of the 41EDO scale.

Some will come closer to JI in some "keys", as I think the one below does.

These can be explored within the same fingering patterns. So one is not starting from scratch each time.

I still like a slightly wide 3/2 plus a few cents, and a slightly wide octave.

Different variations sound different in different keys, and have different beatings.

I don't like too severe a beating, but a little feels good. (Cue Percy Grainger jokes.)

41 feels like the best framework to settle on.

The version below gets 16/15, 8/7, 7/6, 11/8 pretty darn close.

The octave stretch doesn't bother me at all. (It even sounds good.)

The title is a reference to that great piece of cinema by Mr. Ed Wood--Plan 9 from Outer Space.

Caleb

!234.92 gen in 1202.595 octave
41 near-EDO, (28 cents and 39 cents)
41
0.,
27.995,
55.99,
83.985,
111.98,
150.935,
178.93,
206.925,
234.92,
262.915,
290.91,
318.905,
346.9,
385.855,
413.85,
441.845,
469.84,
497.835,
525.83,
553.825,
581.82,
!
620.775,
648.77,
676.765,
704.76,
732.755,
760.75,
788.745,
816.74,
855.695,
883.69,
911.685,
939.68,
967.675,
995.67,
1023.665,
1051.66,
1090.615,
1118.61,
1146.605,
1174.6
1202.595

On Sep 11, 2010, at 2:02 PM, Mike Battaglia wrote:

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

Oh, I don't know what I'm doing at your level.

I was just doing trial-and-error deviations from any of the 41 possible generators in 41EDO--basically every pitch and near-miss.

Then I was trying to stretch the octave a little either way to see what would happen.

The idea with the tuning I posted was that it had a pretty good 9/8, 8/7, 7/6 in near-41EDO.

The stretched octave doesn't sound bad to me at all.

Thinking about this, I simply lack a theory of accuracy--some beating sounds good, but with 7's 11's, I like to hear them really close.

I'm not sure how to think about that.

Here's the map for Rodan.

I'll mess around with it.

Rodan

Equal Temperament Mappings
2 3 5 7 11
[< 41 65 95 115 142 ]
< 87 138 202 244 301 ]>

Reduced Mapping
2 3 5 7 11
[< 1 1 -1 3 6 ]
< 0 3 17 -1 -13 ]>

Generator Tunings (cents)
[1200.057, 234.470>

Step Tunings (cents)
[2.091, 12.809>

On Sep 12, 2010, at 1:56 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@> wrote:
> >
> > > !234.92 gen in 1202.595 octave
> > > 41 near-EDO, (28 cents and 39 cents)
> >
> > This is rodan, I temperament I've been working with lately.
> >
>
> I don't get why you want these extreme (relative to the accuracy of the temperament) octave stretches; octave stretching to that extent introduces tuning problems that weren't there originally. Of course maybe you want to sound like a gamelan orchestra, and rodan is a gamelismic temperament. If you stick 41 and 87 into Graham's magic box
>
> http://x31eq.com/temper/net.html
>
> for various limits, you will see it doesn't suggest there is much payoff to tweaking the octave in rodan.
>
>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

Here's Rodan that someone else generated, which I had overlooked.

Probably you, GWS, did this one.

I like it.

It's very close to what I posted except toward the top, where my pitches trend about 5 cents sharper.

! rodan41.scl
Rodan[41] in 128-tET tuning
41
!0.
28.125
56.250
84.375
112.500
150.000
178.125
206.250
234.375
262.500
290.625
318.750
346.875
384.375
412.500
440.625
468.750
496.875
525.000
553.125
581.250
!
618.750
646.875
675.000
703.125
731.250
759.375
787.500
815.625
853.125
881.250
909.375
937.500
965.625
993.750
1021.875
1050.000
1087.500
1115.625
1143.750
1171.875
2/1

On Sep 12, 2010, at 2:18 PM, caleb morgan wrote:

> Oh, I don't know what I'm doing at your level.
>
>
> I was just doing trial-and-error deviations from any of the 41 possible generators in 41EDO--basically every pitch and near-miss.
>
> Then I was trying to stretch the octave a little either way to see what would happen.
>
> The idea with the tuning I posted was that it had a pretty good 9/8, 8/7, 7/6 in near-41EDO.
>
> The stretched octave doesn't sound bad to me at all.
>
> Thinking about this, I simply lack a theory of accuracy--some beating sounds good, but with 7's 11's, I like to hear them really close.
>
> I'm not sure how to think about that.
>
> Here's the map for Rodan.
>
> I'll mess around with it.
>
> Rodan
>
> Equal Temperament Mappings
> 2 3 5 7 11
> [< 41 65 95 115 142 ]
> < 87 138 202 244 301 ]>
>
> Reduced Mapping
> 2 3 5 7 11
> [< 1 1 -1 3 6 ]
> < 0 3 17 -1 -13 ]>
>
> Generator Tunings (cents)
> [1200.057, 234.470>
>
> Step Tunings (cents)
> [2.091, 12.809>
>
>
>
> On Sep 12, 2010, at 1:56 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>> >
>> >
>> >
>> > --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@> wrote:
>> >
>> > > !234.92 gen in 1202.595 octave
>> > > 41 near-EDO, (28 cents and 39 cents)
>> >
>> > This is rodan, I temperament I've been working with lately.
>> >
>>
>> I don't get why you want these extreme (relative to the accuracy of the temperament) octave stretches; octave stretching to that extent introduces tuning problems that weren't there originally. Of course maybe you want to sound like a gamelan orchestra, and rodan is a gamelismic temperament. If you stick 41 and 87 into Graham's magic box
>>
>> http://x31eq.com/temper/net.html
>>
>> for various limits, you will see it doesn't suggest there is much payoff to tweaking the octave in rodan.
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

🔗Graham Breed <gbreed@...>

On 13 September 2010 03:32, caleb morgan <calebmrgn@...> wrote:

>
>
> Ok, I've been happily typing 41 & Some Big N into the magic box, with 11
> and 13-limits, and I note that the generators all fall close to some member
> of 41EDO, and the octaves are a little big or a little small.
>

They'll be close to 41 EDO because the aim is to get close to JI, and 41 EDO
is close to JI.

> Three questions:
>
> Where do the names come from? (not a very important question)
>

I collect them. Most came from the tuning lists at some point. Some are
from Paul Erlich's key paper, some from Gene's old website, some from the
wiki, and some I made up myself.

> How does this program arrive at optimums? (slightly more interesting
> question, but not too important)
>

It does a Tenney-weighted least squares optimization, using a pure Python
linear algebra library.

> Most important: What are some slightly more radical things to try typing
> in, if any--that is, as you said, *less* accuracy, for the experience of
> seeing what happens?
>

You can choose the accuracy you want from the searches.

> (I'm thinking in terms of radically uneven 41EDO)
>

If you want uneven 41 note scales you can pair 41 with your favourite
nowhere-near-JI octave divisions. (If it's uneven, it isn't an EDO.)

Graham

🔗caleb morgan <calebmrgn@...>

Graham wrote:

> If you want uneven 41 note scales you can pair 41 with your favourite nowhere-near-JI octave divisions. (If it's uneven, it isn't an EDO.)

I think I'm exploring 41-ism, rather than only 41EDO. 41-ism just means having 41 pitches ranked from lower to higher, usually--but not exclusively--within an octave.

I'm also going to check out 41 within 3:1, etc.

There could be a 41 which contains approximations of all the far-from-JI EDOs, as you suggest.

There could also be a 41-pitch scale not too far from 41EDO where certain tonalities are very much in tune, others are not usable at all. But one wouldn't have to learn an entirely new geography--just as people bend a 12-pitch scale without having to learn a whole new geography.

With practice, 41-ism will become a framework for hand-habits.

So 41-ism is a framework just as 12-ism is a framework for a lot of people, including the makers of Logic.

caleb

On Sep 12, 2010, at 8:38 PM, Graham Breed wrote:

>
> On 13 September 2010 03:32, caleb morgan <calebmrgn@yahoo.com> wrote:
>
>
> Ok, I've been happily typing 41 & Some Big N into the magic box, with 11 and 13-limits, and I note that the generators all fall close to some member of 41EDO, and the octaves are a little big or a little small.
>
> They'll be close to 41 EDO because the aim is to get close to JI, and 41 EDO is close to JI.
>
> Three questions:
>
> Where do the names come from? (not a very important question)
>
> I collect them. Most came from the tuning lists at some point. Some are from Paul Erlich's key paper, some from Gene's old website, some from the wiki, and some I made up myself.
>
> How does this program arrive at optimums? (slightly more interesting question, but not too important)
>
> It does a Tenney-weighted least squares optimization, using a pure Python linear algebra library.
>
> Most important: What are some slightly more radical things to try typing in, if any--that is, as you said, *less* accuracy, for the experience of seeing what happens?
>
> You can choose the accuracy you want from the searches.
>
> (I'm thinking in terms of radically uneven 41EDO)
>
> If you want uneven 41 note scales you can pair 41 with your favourite nowhere-near-JI octave divisions. (If it's uneven, it isn't an EDO.)
>
>
> Graham
>
>
>

🔗caleb morgan <calebmrgn@...>

I really should read the LMSO manual, because it was only *today* that I discovered the Interactive Scale Quantize feature,

that lets you gradually or partly bend or morph any JI scale to any EDO and vice-versa.

So, this is more fun than a barrel of monkeys.

I can take my 41-tone JI scale and gradually smooth off the rough edges.

Can take any 41 MOS scale like Rodan, etc. and make it a little more JI.

Can take any other EDO and make a 41-note scale have some of its flavor, in an extremely radical or subtle way as I choose.

Just when I thought my experimentations was drawing to a close.

What fun.

caleb

On Sep 13, 2010, at 6:40 AM, caleb morgan wrote:

>
> Graham wrote:
>
>> If you want uneven 41 note scales you can pair 41 with your favourite nowhere-near-JI octave divisions. (If it's uneven, it isn't an EDO.)
>
>
> I think I'm exploring 41-ism, rather than only 41EDO. 41-ism just means having 41 pitches ranked from lower to higher, usually--but not exclusively--within an octave.
>
> I'm also going to check out 41 within 3:1, etc.
>
> There could be a 41 which contains approximations of all the far-from-JI EDOs, as you suggest.
>
> There could also be a 41-pitch scale not too far from 41EDO where certain tonalities are very much in tune, others are not usable at all. But one wouldn't have to learn an entirely new geography--just as people bend a 12-pitch scale without having to learn a whole new geography.
>
> With practice, 41-ism will become a framework for hand-habits.
>
> So 41-ism is a framework just as 12-ism is a framework for a lot of people, including the makers of Logic.
>
> caleb
>
>
>
>
>
> On Sep 12, 2010, at 8:38 PM, Graham Breed wrote:
>
>>
>>
>> On 13 September 2010 03:32, caleb morgan <calebmrgn@...> wrote:
>>
>>
>> Ok, I've been happily typing 41 & Some Big N into the magic box, with 11 and 13-limits, and I note that the generators all fall close to some member of 41EDO, and the octaves are a little big or a little small.
>>
>> They'll be close to 41 EDO because the aim is to get close to JI, and 41 EDO is close to JI.
>>
>> Three questions:
>>
>> Where do the names come from? (not a very important question)
>>
>> I collect them. Most came from the tuning lists at some point. Some are from Paul Erlich's key paper, some from Gene's old website, some from the wiki, and some I made up myself.
>>
>> How does this program arrive at optimums? (slightly more interesting question, but not too important)
>>
>> It does a Tenney-weighted least squares optimization, using a pure Python linear algebra library.
>>
>> Most important: What are some slightly more radical things to try typing in, if any--that is, as you said, *less* accuracy, for the experience of seeing what happens?
>>
>> You can choose the accuracy you want from the searches.
>>
>> (I'm thinking in terms of radically uneven 41EDO)
>>
>> If you want uneven 41 note scales you can pair 41 with your favourite nowhere-near-JI octave divisions. (If it's uneven, it isn't an EDO.)
>>
>>
>> Graham
>>
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

Heh.

Thing is, you're right that I haven't been easily satisfied.

That's why my plans keep changing.

Now, I think I'm settling for uneven fingering and a design much closer to JI, which fits 4 octaves on two 88-key keyboards.

It's very close to the Rube Goldberg scales I came up with, which are basically JI with an attempt at tempering out the difference between 32/27 and 13/11.

At the same time, I discovered that Lil' Miss Scale Oven will let me do compromise tunings between a given scale and some EDO.

My scale is printed below. I tried morphing it 50% if the way to 76EDO, just to see what would happen. The results aren't bad.

What might be some other EDOs to try to "quantize" it, all or part of the way to?

Caleb

43-pitch 13 Rube Goldberg
43
!cents PC Approximate ratio
!0. 0 1/1
84.5 1 [4/3 below 7/5, or 21/20]
117.0 2 16/15-ish
135.0 3 13/12 was 138.6
150.6 4 12/11
!
165.0 5 11/10
179.1 6 10/9 was 182.4
207.2 7 9/8 wide with 3/2
231.2 8 8/7
247.74 9 15/13
265.2 10 7/6 lowered for low 4/3
!
289.2 11 13/11 and tempered 32/27
313.6 0 6/5
344.1 1 11/9 tempered
359.47 2 16/13
385.0 3 5/4
414.5 4 14/11 low to go with 22/13
!
435.1 5 9/7
496.4 6 4/3 low
551.3 7 11/8
563.4 8 18/13
582.5 9 7/5
!
617.5 10 10/7
648.7 11 16/11
703.6 0 3/2 wide
745.8 1 20/13
!
764.9 2 14/9
782.5 3 11/7
819.0 4 8/5 high, originally 813.78
840.53 5 13/8
852.6 6 18/11
882.7 7 5/3 lowered with 4/3
!
910.79 8 22/13
933.1 9 12/7
968.8 10 7/4
992.8 11 16/9 low with 4/3
1013.6 0 9/5
!
1035.0 1 20/11
1049.4 2 11/6
1061.4 3 24/13
1071.7 4 13/7
1085.0 5 15/8
1115.5 6 [4/3 above 10/7, or 40/21]
!
1200.0 0 2/1

Attempt to quantize to nearest 72EDO by 50% percent

0., 83.917, 116.833, 134.167, 150.3, 165.833, 181.217, 203.6, 232.267, 248.87, 265.933, 286.267, 315.133, 347.051, 363.068, 384.167, 415.583, 434.217, 498.2, 550.65, 565.033, 582.917, 617.083, 649.35, 701.8, 747.9, 765.783, 782.917, 817.833, 836.932, 851.3, 883.017, 913.728, 933.217, 967.733, 996.401, 1015.133, 1034.167, 1049.7, 1064.033, 1069.183, 1084.167, 1116.083

On Sep 11, 2010, at 2:13 AM, jonszanto wrote:

> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
> > It might look like a slight tempering of Partch's scale.
> >
> > This is my current thinking.
> >
> > I'm quite serious.
>
> Compromise is for pussies. In your heart, you know it's true.
>
>

🔗caleb morgan <calebmrgn@...>

oops, meant 72EDO, not 76.

-c

On Sep 14, 2010, at 11:27 AM, caleb morgan wrote:

>
> Heh.
>
> Thing is, you're right that I haven't been easily satisfied.
>
> That's why my plans keep changing.
>
> Now, I think I'm settling for uneven fingering and a design much closer to JI, which fits 4 octaves on two 88-key keyboards.
>
> It's very close to the Rube Goldberg scales I came up with, which are basically JI with an attempt at tempering out the difference between 32/27 and 13/11.
>
> At the same time, I discovered that Lil' Miss Scale Oven will let me do compromise tunings between a given scale and some EDO.
>
> My scale is printed below. I tried morphing it 50% if the way to 76EDO, just to see what would happen. The results aren't bad.
>
> What might be some other EDOs to try to "quantize" it, all or part of the way to?
>
> Caleb
>
> 43-pitch 13 Rube Goldberg
> 43
> !cents PC Approximate ratio
> !0. 0 1/1
> 84.5 1 [4/3 below 7/5, or 21/20]
> 117.0 2 16/15-ish
> 135.0 3 13/12 was 138.6
> 150.6 4 12/11
> !
> 165.0 5 11/10
> 179.1 6 10/9 was 182.4
> 207.2 7 9/8 wide with 3/2
> 231.2 8 8/7
> 247.74 9 15/13
> 265.2 10 7/6 lowered for low 4/3
> !
> 289.2 11 13/11 and tempered 32/27
> 313.6 0 6/5
> 344.1 1 11/9 tempered
> 359.47 2 16/13
> 385.0 3 5/4
> 414.5 4 14/11 low to go with 22/13
> !
> 435.1 5 9/7
> 496.4 6 4/3 low
> 551.3 7 11/8
> 563.4 8 18/13
> 582.5 9 7/5
> !
> 617.5 10 10/7
> 648.7 11 16/11
> 703.6 0 3/2 wide
> 745.8 1 20/13
> !
> 764.9 2 14/9
> 782.5 3 11/7
> 819.0 4 8/5 high, originally 813.78
> 840.53 5 13/8
> 852.6 6 18/11
> 882.7 7 5/3 lowered with 4/3
> !
> 910.79 8 22/13
> 933.1 9 12/7
> 968.8 10 7/4
> 992.8 11 16/9 low with 4/3
> 1013.6 0 9/5
> !
> 1035.0 1 20/11
> 1049.4 2 11/6
> 1061.4 3 24/13
> 1071.7 4 13/7
> 1085.0 5 15/8
> 1115.5 6 [4/3 above 10/7, or 40/21]
> !
> 1200.0 0 2/1
>
>
> Attempt to quantize to nearest 72EDO by 50% percent
>
> 0., 83.917, 116.833, 134.167, 150.3, 165.833, 181.217, 203.6, 232.267, 248.87, 265.933, 286.267, 315.133, 347.051, 363.068, 384.167, 415.583, 434.217, 498.2, 550.65, 565.033, 582.917, 617.083, 649.35, 701.8, 747.9, 765.783, 782.917, 817.833, 836.932, 851.3, 883.017, 913.728, 933.217, 967.733, 996.401, 1015.133, 1034.167, 1049.7, 1064.033, 1069.183, 1084.167, 1116.083
>
>
>
>
> On Sep 11, 2010, at 2:13 AM, jonszanto wrote:
>
>>
>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>> > It might look like a slight tempering of Partch's scale.
>> >
>> > This is my current thinking.
>> >
>> > I'm quite serious.
>>
>> Compromise is for pussies. In your heart, you know it's true.
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

Here I looked up EDOs on the Xenharmony page and chose 87EDO to quantize to:

0., 82.759, 110.345, 137.931, 151.724, 165.517, 179.31, 206.897, 234.483, 248.276, 262.069, 289.655, 317.241, 344.828, 358.621, 386.207, 413.793, 441.379, 496.552, 551.724, 565.517, 579.31, 620.69, 648.276, 703.448, 744.828, 758.621, 786.207, 813.793, 841.379, 855.172, 882.759, 910.345, 937.931, 965.517, 993.103, 1006.897, 1034.483, 1048.276, 1062.069, 1075.862, 1089.655, 1117.241

looks pretty good!

I'm not sure about advantages or disadvantages, yet.

It makes all the step sizes multiples of 13.793 cents, with a "sruti" pattern of:

6, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 4, 1, 1, 3, 2, 4, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 2, 6

-c

On Sep 14, 2010, at 11:34 AM, caleb morgan wrote:

> oops, meant 72EDO, not 76.
>
>
> -c
>
>
> On Sep 14, 2010, at 11:27 AM, caleb morgan wrote:
>
>>
>>
>> Heh.
>>
>> Thing is, you're right that I haven't been easily satisfied.
>>
>> That's why my plans keep changing.
>>
>> Now, I think I'm settling for uneven fingering and a design much closer to JI, which fits 4 octaves on two 88-key keyboards.
>>
>> It's very close to the Rube Goldberg scales I came up with, which are basically JI with an attempt at tempering out the difference between 32/27 and 13/11.
>>
>> At the same time, I discovered that Lil' Miss Scale Oven will let me do compromise tunings between a given scale and some EDO.
>>
>> My scale is printed below. I tried morphing it 50% if the way to 76EDO, just to see what would happen. The results aren't bad.
>>
>> What might be some other EDOs to try to "quantize" it, all or part of the way to?
>>
>> Caleb
>>
>> 43-pitch 13 Rube Goldberg
>> 43
>> !cents PC Approximate ratio
>> !0. 0 1/1
>> 84.5 1 [4/3 below 7/5, or 21/20]
>> 117.0 2 16/15-ish
>> 135.0 3 13/12 was 138.6
>> 150.6 4 12/11
>> !
>> 165.0 5 11/10
>> 179.1 6 10/9 was 182.4
>> 207.2 7 9/8 wide with 3/2
>> 231.2 8 8/7
>> 247.74 9 15/13
>> 265.2 10 7/6 lowered for low 4/3
>> !
>> 289.2 11 13/11 and tempered 32/27
>> 313.6 0 6/5
>> 344.1 1 11/9 tempered
>> 359.47 2 16/13
>> 385.0 3 5/4
>> 414.5 4 14/11 low to go with 22/13
>> !
>> 435.1 5 9/7
>> 496.4 6 4/3 low
>> 551.3 7 11/8
>> 563.4 8 18/13
>> 582.5 9 7/5
>> !
>> 617.5 10 10/7
>> 648.7 11 16/11
>> 703.6 0 3/2 wide
>> 745.8 1 20/13
>> !
>> 764.9 2 14/9
>> 782.5 3 11/7
>> 819.0 4 8/5 high, originally 813.78
>> 840.53 5 13/8
>> 852.6 6 18/11
>> 882.7 7 5/3 lowered with 4/3
>> !
>> 910.79 8 22/13
>> 933.1 9 12/7
>> 968.8 10 7/4
>> 992.8 11 16/9 low with 4/3
>> 1013.6 0 9/5
>> !
>> 1035.0 1 20/11
>> 1049.4 2 11/6
>> 1061.4 3 24/13
>> 1071.7 4 13/7
>> 1085.0 5 15/8
>> 1115.5 6 [4/3 above 10/7, or 40/21]
>> !
>> 1200.0 0 2/1
>>
>>
>> Attempt to quantize to nearest 72EDO by 50% percent
>>
>> 0., 83.917, 116.833, 134.167, 150.3, 165.833, 181.217, 203.6, 232.267, 248.87, 265.933, 286.267, 315.133, 347.051, 363.068, 384.167, 415.583, 434.217, 498.2, 550.65, 565.033, 582.917, 617.083, 649.35, 701.8, 747.9, 765.783, 782.917, 817.833, 836.932, 851.3, 883.017, 913.728, 933.217, 967.733, 996.401, 1015.133, 1034.167, 1049.7, 1064.033, 1069.183, 1084.167, 1116.083
>>
>>
>>
>>
>> On Sep 11, 2010, at 2:13 AM, jonszanto wrote:
>>
>>>
>>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>> > It might look like a slight tempering of Partch's scale.
>>> >
>>> > This is my current thinking.
>>> >
>>> > I'm quite serious.
>>>
>>> Compromise is for pussies. In your heart, you know it's true.
>>>
>>
>>
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

Just a brief summary of the method to my madness.

Without being able, at the moment, to put my finger on exactly *what*, I've felt something missing from 41EDO and variations on it.

I tried some crazy experiments with it, trying to stick pitches I wanted in spots in mod 41 where I would never use those pitches--scale degrees 2,3, 39, and 40. There were a few others, but those were the obvious candidates. The results were absurd at best.

I like some aspects of 46 EDO, but I can't fit two octaves on a standard 88-note keyboard. I prefer the 10/9, 9/8, 8/7, 7/6 approximation in 41EDO anyway.

Research into bigger standard keyboards produced zip, nada, bupkis.

With only 88 keys, you get the feeling that you don't want to waste any of them, not even 4.

I briefly considered a double (two-octave) scale of 46 EDO with 4 pitches lopped off the top, but that didn't cut it.

So, I decided to accept inconsistent fingering and go for having enough pitches with enough accuracy. I can get more pitches by simply pitch-bending the whole tuning base--I'd already constructed the tables of values. Previous work has shown this to be a somewhat funky but reliable method.

The idea--proposed by someone--that one could work with a subset of the pitches one needs in Logic--which is limited to 12 pitches--and then divide the composition into groups of 12 pitches used--is insupportable because I need to be able to hear the scales changing, and I need to be able to experiment, get the sound in my hands. It's about as impractical an approach as I can possibly imagine.

I'm not ready at this time to tackle dynamic retuning with LMSO, and I'm not sure if it will meet my needs.

I've worked more or less this way--in the past--with a 36-pitch JI scale, by pitch-bending the instruments when going to different tuning-bases. So I know this method will work.

Only the scale remained to be worked out exactly. I was hoping to broaden my horizons a little over 36-pitch JI.

Using Lil' Miss Scale Oven (LMSO), I designed a big JI tonality diamond with a 19-limit. Then I threw out pitches to bring it down first to 60, then 48, then 43 pitches. Sadly, the 19s and 17s had to go. Some of the 15s had to be cheated a little, and some other pitches had to be tempered out.

43 is the most I can really fit.

Then using the Quantize feature of LMSO, I've made my scale conform to 72, 87, 94. and 104 EDO, because the brilliant folks here have already done the research and understand what EDOs have good conformation to JI.

So, I at least have a framework, with a continuum from almost-pure JI to something with a little more of a tempered sound.

It seems to work--for me at least.

It's a little hard to explain why I can tolerate some beating in octaves, some very slight sharpness in the 5ths (3/2s), but want to hear many of the 7's 11's and 13's dead-on. It might be because of the larger "field of attraction" of the very simple ratios.

72, 87, 94, 104 EDOs produce small variations in the sound of the scale--different rates of beating, subtilely different colors.

Caleb

On Sep 14, 2010, at 12:57 PM, caleb morgan wrote:

>
> Here I looked up EDOs on the Xenharmony page and chose 87EDO to quantize to:
>
> 0., 82.759, 110.345, 137.931, 151.724, 165.517, 179.31, 206.897, 234.483, 248.276, 262.069, 289.655, 317.241, 344.828, 358.621, 386.207, 413.793, 441.379, 496.552, 551.724, 565.517, 579.31, 620.69, 648.276, 703.448, 744.828, 758.621, 786.207, 813.793, 841.379, 855.172, 882.759, 910.345, 937.931, 965.517, 993.103, 1006.897, 1034.483, 1048.276, 1062.069, 1075.862, 1089.655, 1117.241
>
> looks pretty good!
>
> I'm not sure about advantages or disadvantages, yet.
>
> It makes all the step sizes multiples of 13.793 cents, with a "sruti" pattern of:
>
> 6, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 4, 1, 1, 3, 2, 4, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 2, 6
>
> -c
>
>
>
> On Sep 14, 2010, at 11:34 AM, caleb morgan wrote:
>
>>
>> oops, meant 72EDO, not 76.
>>
>>
>> -c
>>
>>
>> On Sep 14, 2010, at 11:27 AM, caleb morgan wrote:
>>
>>>
>>>
>>> Heh.
>>>
>>> Thing is, you're right that I haven't been easily satisfied.
>>>
>>> That's why my plans keep changing.
>>>
>>> Now, I think I'm settling for uneven fingering and a design much closer to JI, which fits 4 octaves on two 88-key keyboards.
>>>
>>> It's very close to the Rube Goldberg scales I came up with, which are basically JI with an attempt at tempering out the difference between 32/27 and 13/11.
>>>
>>> At the same time, I discovered that Lil' Miss Scale Oven will let me do compromise tunings between a given scale and some EDO.
>>>
>>> My scale is printed below. I tried morphing it 50% if the way to 76EDO, just to see what would happen. The results aren't bad.
>>>
>>> What might be some other EDOs to try to "quantize" it, all or part of the way to?
>>>
>>> Caleb
>>>
>>> 43-pitch 13 Rube Goldberg
>>> 43
>>> !cents PC Approximate ratio
>>> !0. 0 1/1
>>> 84.5 1 [4/3 below 7/5, or 21/20]
>>> 117.0 2 16/15-ish
>>> 135.0 3 13/12 was 138.6
>>> 150.6 4 12/11
>>> !
>>> 165.0 5 11/10
>>> 179.1 6 10/9 was 182.4
>>> 207.2 7 9/8 wide with 3/2
>>> 231.2 8 8/7
>>> 247.74 9 15/13
>>> 265.2 10 7/6 lowered for low 4/3
>>> !
>>> 289.2 11 13/11 and tempered 32/27
>>> 313.6 0 6/5
>>> 344.1 1 11/9 tempered
>>> 359.47 2 16/13
>>> 385.0 3 5/4
>>> 414.5 4 14/11 low to go with 22/13
>>> !
>>> 435.1 5 9/7
>>> 496.4 6 4/3 low
>>> 551.3 7 11/8
>>> 563.4 8 18/13
>>> 582.5 9 7/5
>>> !
>>> 617.5 10 10/7
>>> 648.7 11 16/11
>>> 703.6 0 3/2 wide
>>> 745.8 1 20/13
>>> !
>>> 764.9 2 14/9
>>> 782.5 3 11/7
>>> 819.0 4 8/5 high, originally 813.78
>>> 840.53 5 13/8
>>> 852.6 6 18/11
>>> 882.7 7 5/3 lowered with 4/3
>>> !
>>> 910.79 8 22/13
>>> 933.1 9 12/7
>>> 968.8 10 7/4
>>> 992.8 11 16/9 low with 4/3
>>> 1013.6 0 9/5
>>> !
>>> 1035.0 1 20/11
>>> 1049.4 2 11/6
>>> 1061.4 3 24/13
>>> 1071.7 4 13/7
>>> 1085.0 5 15/8
>>> 1115.5 6 [4/3 above 10/7, or 40/21]
>>> !
>>> 1200.0 0 2/1
>>>
>>>
>>> Attempt to quantize to nearest 72EDO by 50% percent
>>>
>>> 0., 83.917, 116.833, 134.167, 150.3, 165.833, 181.217, 203.6, 232.267, 248.87, 265.933, 286.267, 315.133, 347.051, 363.068, 384.167, 415.583, 434.217, 498.2, 550.65, 565.033, 582.917, 617.083, 649.35, 701.8, 747.9, 765.783, 782.917, 817.833, 836.932, 851.3, 883.017, 913.728, 933.217, 967.733, 996.401, 1015.133, 1034.167, 1049.7, 1064.033, 1069.183, 1084.167, 1116.083
>>>
>>>
>>>
>>>
>>> On Sep 11, 2010, at 2:13 AM, jonszanto wrote:
>>>
>>>>
>>>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>>> > It might look like a slight tempering of Partch's scale.
>>>> >
>>>> > This is my current thinking.
>>>> >
>>>> > I'm quite serious.
>>>>
>>>> Compromise is for pussies. In your heart, you know it's true.
>>>>
>>>
>>>
>>
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

http://www.box.net/shared/m37jhti1og#/shared/m37jhti1og/1/51261090

For those who want to check out my "all the pitch that fits" approach, I've posted the original 43-note Rube Goldberg (RG) file,

plus 15 variations--two are more in tune with JI, and the other 13 or so are nearly or exactly in the following EDOs:

46, 53, 72, 94, 96, 104, 159, 166, 171, 185, 190, 205,

I chose these, on the whole, to make the 3/2 and the 9/8, if anything, less sharp, and I chose EDOs that didn't eliminate any of the 43 pitches.

Somewhat to my surprise, they actually sound different in some intervals.

The idea for this EDOs came from the Xenharmonic page:

http://xenharmonic.wikispaces.com/edo

Which is incredibly helpful.

So, I thank Graham Breed, Gene Ward Smith, Margo Schulter, Ozan Yarman, George Secor, for their work. (Even If I accomplish nothing, I'm still appreciative.)

ah, heck, even Carl Lumma and Mike Battaglia.

They, more than the Flying Spaghetti Monster, have really helped me. But they bear no responsibility for my folly.

Caleb

On Sep 14, 2010, at 3:08 PM, caleb morgan wrote:

>
> Just a brief summary of the method to my madness.
>
> Without being able, at the moment, to put my finger on exactly *what*, I've felt something missing from 41EDO and variations on it.
>
> I tried some crazy experiments with it, trying to stick pitches I wanted in spots in mod 41 where I would never use those pitches--scale degrees 2,3, 39, and 40. There were a few others, but those were the obvious candidates. The results were absurd at best.
>
> I like some aspects of 46 EDO, but I can't fit two octaves on a standard 88-note keyboard. I prefer the 10/9, 9/8, 8/7, 7/6 approximation in 41EDO anyway.
>
> Research into bigger standard keyboards produced zip, nada, bupkis.
>
> With only 88 keys, you get the feeling that you don't want to waste any of them, not even 4.
>
> I briefly considered a double (two-octave) scale of 46 EDO with 4 pitches lopped off the top, but that didn't cut it.
>
> So, I decided to accept inconsistent fingering and go for having enough pitches with enough accuracy. I can get more pitches by simply pitch-bending the whole tuning base--I'd already constructed the tables of values. Previous work has shown this to be a somewhat funky but reliable method.
>
> The idea--proposed by someone--that one could work with a subset of the pitches one needs in Logic--which is limited to 12 pitches--and then divide the composition into groups of 12 pitches used--is insupportable because I need to be able to hear the scales changing, and I need to be able to experiment, get the sound in my hands. It's about as impractical an approach as I can possibly imagine.
>
> I'm not ready at this time to tackle dynamic retuning with LMSO, and I'm not sure if it will meet my needs.
>
> I've worked more or less this way--in the past--with a 36-pitch JI scale, by pitch-bending the instruments when going to different tuning-bases. So I know this method will work.
>
> Only the scale remained to be worked out exactly. I was hoping to broaden my horizons a little over 36-pitch JI.
>
> Using Lil' Miss Scale Oven (LMSO), I designed a big JI tonality diamond with a 19-limit. Then I threw out pitches to bring it down first to 60, then 48, then 43 pitches. Sadly, the 19s and 17s had to go. Some of the 15s had to be cheated a little, and some other pitches had to be tempered out.
>
> 43 is the most I can really fit.
>
> Then using the Quantize feature of LMSO, I've made my scale conform to 72, 87, 94. and 104 EDO, because the brilliant folks here have already done the research and understand what EDOs have good conformation to JI.
>
> So, I at least have a framework, with a continuum from almost-pure JI to something with a little more of a tempered sound.
>
> It seems to work--for me at least.
>
> It's a little hard to explain why I can tolerate some beating in octaves, some very slight sharpness in the 5ths (3/2s), but want to hear many of the 7's 11's and 13's dead-on. It might be because of the larger "field of attraction" of the very simple ratios.
>
> 72, 87, 94, 104 EDOs produce small variations in the sound of the scale--different rates of beating, subtilely different colors.
>
> Caleb
>
>
>
>
>
>
>
>
>
>
>
> On Sep 14, 2010, at 12:57 PM, caleb morgan wrote:
>
>>
>>
>> Here I looked up EDOs on the Xenharmony page and chose 87EDO to quantize to:
>>
>> 0., 82.759, 110.345, 137.931, 151.724, 165.517, 179.31, 206.897, 234.483, 248.276, 262.069, 289.655, 317.241, 344.828, 358.621, 386.207, 413.793, 441.379, 496.552, 551.724, 565.517, 579.31, 620.69, 648.276, 703.448, 744.828, 758.621, 786.207, 813.793, 841.379, 855.172, 882.759, 910.345, 937.931, 965.517, 993.103, 1006.897, 1034.483, 1048.276, 1062.069, 1075.862, 1089.655, 1117.241
>>
>> looks pretty good!
>>
>> I'm not sure about advantages or disadvantages, yet.
>>
>> It makes all the step sizes multiples of 13.793 cents, with a "sruti" pattern of:
>>
>> 6, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 4, 1, 1, 3, 2, 4, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 1, 1, 2, 6
>>
>> -c
>>
>>
>>
>> On Sep 14, 2010, at 11:34 AM, caleb morgan wrote:
>>
>>>
>>> oops, meant 72EDO, not 76.
>>>
>>>
>>> -c
>>>
>>>
>>> On Sep 14, 2010, at 11:27 AM, caleb morgan wrote:
>>>
>>>>
>>>>
>>>> Heh.
>>>>
>>>> Thing is, you're right that I haven't been easily satisfied.
>>>>
>>>> That's why my plans keep changing.
>>>>
>>>> Now, I think I'm settling for uneven fingering and a design much closer to JI, which fits 4 octaves on two 88-key keyboards.
>>>>
>>>> It's very close to the Rube Goldberg scales I came up with, which are basically JI with an attempt at tempering out the difference between 32/27 and 13/11.
>>>>
>>>> At the same time, I discovered that Lil' Miss Scale Oven will let me do compromise tunings between a given scale and some EDO.
>>>>
>>>> My scale is printed below. I tried morphing it 50% if the way to 76EDO, just to see what would happen. The results aren't bad.
>>>>
>>>> What might be some other EDOs to try to "quantize" it, all or part of the way to?
>>>>
>>>> Caleb
>>>>
>>>> 43-pitch 13 Rube Goldberg
>>>> 43
>>>> !cents PC Approximate ratio
>>>> !0. 0 1/1
>>>> 84.5 1 [4/3 below 7/5, or 21/20]
>>>> 117.0 2 16/15-ish
>>>> 135.0 3 13/12 was 138.6
>>>> 150.6 4 12/11
>>>> !
>>>> 165.0 5 11/10
>>>> 179.1 6 10/9 was 182.4
>>>> 207.2 7 9/8 wide with 3/2
>>>> 231.2 8 8/7
>>>> 247.74 9 15/13
>>>> 265.2 10 7/6 lowered for low 4/3
>>>> !
>>>> 289.2 11 13/11 and tempered 32/27
>>>> 313.6 0 6/5
>>>> 344.1 1 11/9 tempered
>>>> 359.47 2 16/13
>>>> 385.0 3 5/4
>>>> 414.5 4 14/11 low to go with 22/13
>>>> !
>>>> 435.1 5 9/7
>>>> 496.4 6 4/3 low
>>>> 551.3 7 11/8
>>>> 563.4 8 18/13
>>>> 582.5 9 7/5
>>>> !
>>>> 617.5 10 10/7
>>>> 648.7 11 16/11
>>>> 703.6 0 3/2 wide
>>>> 745.8 1 20/13
>>>> !
>>>> 764.9 2 14/9
>>>> 782.5 3 11/7
>>>> 819.0 4 8/5 high, originally 813.78
>>>> 840.53 5 13/8
>>>> 852.6 6 18/11
>>>> 882.7 7 5/3 lowered with 4/3
>>>> !
>>>> 910.79 8 22/13
>>>> 933.1 9 12/7
>>>> 968.8 10 7/4
>>>> 992.8 11 16/9 low with 4/3
>>>> 1013.6 0 9/5
>>>> !
>>>> 1035.0 1 20/11
>>>> 1049.4 2 11/6
>>>> 1061.4 3 24/13
>>>> 1071.7 4 13/7
>>>> 1085.0 5 15/8
>>>> 1115.5 6 [4/3 above 10/7, or 40/21]
>>>> !
>>>> 1200.0 0 2/1
>>>>
>>>>
>>>> Attempt to quantize to nearest 72EDO by 50% percent
>>>>
>>>> 0., 83.917, 116.833, 134.167, 150.3, 165.833, 181.217, 203.6, 232.267, 248.87, 265.933, 286.267, 315.133, 347.051, 363.068, 384.167, 415.583, 434.217, 498.2, 550.65, 565.033, 582.917, 617.083, 649.35, 701.8, 747.9, 765.783, 782.917, 817.833, 836.932, 851.3, 883.017, 913.728, 933.217, 967.733, 996.401, 1015.133, 1034.167, 1049.7, 1064.033, 1069.183, 1084.167, 1116.083
>>>>
>>>>
>>>>
>>>>
>>>> On Sep 11, 2010, at 2:13 AM, jonszanto wrote:
>>>>
>>>>>
>>>>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>>>> > It might look like a slight tempering of Partch's scale.
>>>>> >
>>>>> > This is my current thinking.
>>>>> >
>>>>> > I'm quite serious.
>>>>>
>>>>> Compromise is for pussies. In your heart, you know it's true.
>>>>>
>>>>
>>>>
>>>
>>>
>>
>>
>
>
>

🔗caleb morgan <calebmrgn@...>

I wanted to make this "something missing" point a little more substantial.

Specifically, in 41EDO I miss:

13/12 @ 138.57 cents

11/10 @ 165 cents

15/13 @ 247.7 cents

11/9 @ 347.4 cents

and so forth.

So, a lot of 11 and 13 harmony.

While this fact is hardly a secret, it bears repeating for people like me--rather than for the old hands and experts here.

Also, the 5/4 feels slightly low, though not bad.

The list continues, but I wanted to confirm to myself and anyone else interested in 11 and 13-ratio harmonic sounds that these aren't well-represented.

Plus, it has, for my purposes, 4 pitches I would never use right next to 1/1 and 2/1: 29.26, 58.54, 1141.4, and 1170.7.

So, as Mr. Lumma said, 41 is not bad for an EDO. It's darn good.

I wonder, for my own scale design--before I attempt to "burn in" my current scale--whether I ought to attempt something like Mod43-consistent 5ths and 4ths.

A helpful tool for consistent fingering is this modular multiplier:

http://pages.central.edu/emp/LintonT/classes/spring01/cryptography/java/Multiples.html

But I think that any scheme will have to cheat something, and I've chosen to cheat regular fingering, which I intend to get around by simply labelling
my keyboard with the ratios, written on a long swatch of tape above the keys, just like mixing boards are labelled with the instruments.

I've already done this, and it makes finding the pitches pretty easy.

The scale I intend to practice is as evenly proportioned as the sensory-motor homunculus mapping in the cortex. (That is, it's a little screwy.)

http://www.google.com/images?hl=en&expIds=17259,18168,25260,25900,26447,26515,26565&sugexp=ldymls&xhr=t&q=sensory-motor+homunculus&cp=14&um=1&ie=UTF-8&source=og&sa=N&tab=wi&biw=1065&bih=836

If anyone has any (non-abusive) thoughts, let me know.

Perhaps there's a mass-produced 100+ key Midi controller.

That might change everything.

Caleb

On Sep 14, 2010, at 3:08 PM, caleb morgan wrote:

>
> Just a brief summary of the method to my madness.
>
> Without being able, at the moment, to put my finger on exactly *what*, I've felt something missing from 41EDO and variations on it.
>

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

This scale seems to be the clear winner.

I have nothing to say, so I'm going to post some doggerel.

Tuning and tuning in his widening gyre,
The Fatso tries an EDO.

Fifths move apart, the tritone cannot hold;
The Fatso wears a Speedo.

Sure, some tunings fit his hands.
Sure some Miracle has hit the fan,

Another tuning! The tuner's had enough,
But he only sits and eats some Cheetos.

The time to Shed has come again,
his happy tinkering at an end,

Patience, practice, is his credo,
But the fatso lacks libido.

(tuner, fatso = me, so no one should think this is about them.)

caleb

On Sep 16, 2010, at 5:30 AM, caleb morgan wrote:

> Wow. This looks good. You're not making it easy for me to settle on something. I'll play this one for a while too, and if I have anything to say, later, I'll say it. Much of it comes very close to the scale I was going to practice, so I'm hoping some tweak/synthesis of the two might somehow be possible.
>
>
> A lot of this scale comes close to 72EDO, except this looks better.
>
> -c
>
>
> On Sep 15, 2010, at 8:54 PM, genewardsmith wrote:
>
>> ! miracle41s.scl
>> !
>> Miracle-41 with Secor's minimax generator of 116.7155941 cents (5:9 exact). XH5, 1976
>> 41
>> !
>> 32.84406
>> 51.02748
>> 83.87154
>> 116.71559
>> 149.55965
>> 10/9
>> 200.58713
>> 233.43119
>> 266.27525
>> 299.11931
>> 317.30272
>> 350.14678
>> 382.99084
>> 415.83490
>> 434.01832
>> 466.86238
>> 499.70643
>> 532.55049
>> 550.73391
>> 583.57797
>> 616.42203
>> 649.26609
>> 667.44951
>> 700.29357
>> 733.13762
>> 765.98168
>> 784.16510
>> 817.00916
>> 849.85322
>> 882.69728
>> 900.88069
>> 933.72475
>> 966.56881
>> 999.41287
>> 9/5
>> 1050.44035
>> 1083.28441
>> 1116.12847
>> 1134.31188
>> 1167.15594
>> 2/1
>
>
>

🔗caleb morgan <calebmrgn@...>

>all the pitches I want except 22/13 and 20/11? check

>consistent fingering? check

Darn your superior facts, logic and knowledge of scale theory.

But...but...doesn't this here miracle41s scale lack some ineffable um, *soul*? Perhaps it's....*too* perfect. Yeah, that's the ticket.

Do I care that the 5ths are a teeny-tiny bit low?

-c

On Sep 16, 2010, at 5:30 AM, caleb morgan wrote:

🔗caleb morgan <calebmrgn@...>

Here are 4 micro-variations on miracle41 stud-loco,

by quantizing to different EDOs, using Lil Miss Scale Oven.

I just had to try it, I'm not sure if there are any advantages, yet.

But they all do sound a little different.

!
miracle 41 Stud Loco in 46EDO
41
!
26.08700
52.17400
78.26100
104.34800
156.52200
182.60900
208.69600
234.78300
260.87000
286.95700
313.04300
339.13000
391.30400
417.39100
443.47800
469.56500
495.65200
521.73900
547.82600
573.91300
626.08700
652.17400
678.26100
704.34800
730.43500
756.52200
782.60900
808.69600
860.87000
886.95700
913.04300
939.13000
965.21700
991.30400
1017.39100
1043.47800
1095.65200
1121.73900
1147.82600
1173.91300
1200.00000

!
miracle41 SL in 94 EDO
41
!
38.29787
51.06383
89.36170
114.89362
153.19149
178.72340
204.25532
229.78723
268.08511
293.61702
319.14894
344.68085
382.97872
421.27660
434.04255
472.34043
497.87234
536.17021
548.93617
587.23404
612.76596
651.06383
663.82979
702.12766
727.65957
765.95745
778.72340
817.02128
855.31915
880.85106
906.38298
931.91489
970.21277
995.74468
1021.27660
1046.80851
1085.10638
1110.63830
1136.17021
1161.70213
1200.00000

! miracle 41 Stud Loco quantized to 159 EDO
miracle41s in 159 EDO
41
!
30.18900
52.83000
83.01900
113.20800
150.94300
181.13200
203.77400
233.96200
264.15100
301.88700
316.98100
347.17000
384.90600
415.09400
437.73600
467.92500
498.11300
535.84900
550.94300
581.13200
618.86800
649.05700
664.15100
701.88700
732.07500
762.26400
784.90600
815.09400
852.83000
883.01900
898.11300
935.84900
966.03800
996.22600
1018.86800
1049.05700
1086.79200
1116.98100
1132.07600
1169.81100
1200.00000

!
miracle41s in 171 EDO
41
!
35.08800
49.12300
84.21100
119.29800
147.36800
182.45600
203.50900
231.57900
266.66700
301.75400
315.78900
350.87700
385.96500
414.03500
435.08800
470.17500
498.24600
533.33300
547.36800
582.45600
617.54400
652.63200
666.66700
701.75400
729.82500
764.91200
785.96500
814.03500
849.12300
884.21100
898.24600
933.33300
968.42100
996.49100
1017.54400
1052.63200
1080.70200
1115.78900
1136.84200
1164.91200
1200.00000

On Sep 16, 2010, at 6:21 AM, caleb morgan wrote:

> >all the pitches I want except 22/13 and 20/11? check
>
>
> >consistent fingering? check
>
> Darn your superior facts, logic and knowledge of scale theory.
>
> But...but...doesn't this here miracle41s scale lack some ineffable um, *soul*? Perhaps it's....*too* perfect. Yeah, that's the ticket.
>
> Do I care that the 5ths are a teeny-tiny bit low?
>
> -c
>
>
> On Sep 16, 2010, at 5:30 AM, caleb morgan wrote:
>
>>
>> Wow. This looks good. You're not making it easy for me to settle on something. I'll play this one for a while too, and if I have anything to say, later, I'll say it. Much of it comes very close to the scale I was going to practice, so I'm hoping some tweak/synthesis of the two might somehow be possible.
>>
>>
>> A lot of this scale comes close to 72EDO, except this looks better.
>>
>> -c
>>
>>
>> On Sep 15, 2010, at 8:54 PM, genewardsmith wrote:
>>
>>> ! miracle41s.scl
>>> !
>>> Miracle-41 with Secor's minimax generator of 116.7155941 cents (5:9 exact). XH5, 1976
>>> 41
>>> !
>>> 32.84406
>>> 51.02748
>>> 83.87154
>>> 116.71559
>>> 149.55965
>>> 10/9
>>> 200.58713
>>> 233.43119
>>> 266.27525
>>> 299.11931
>>> 317.30272
>>> 350.14678
>>> 382.99084
>>> 415.83490
>>> 434.01832
>>> 466.86238
>>> 499.70643
>>> 532.55049
>>> 550.73391
>>> 583.57797
>>> 616.42203
>>> 649.26609
>>> 667.44951
>>> 700.29357
>>> 733.13762
>>> 765.98168
>>> 784.16510
>>> 817.00916
>>> 849.85322
>>> 882.69728
>>> 900.88069
>>> 933.72475
>>> 966.56881
>>> 999.41287
>>> 9/5
>>> 1050.44035
>>> 1083.28441
>>> 1116.12847
>>> 1134.31188
>>> 1167.15594
>>> 2/1
>>
>>
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

Just to clear up one thing.

When I read the response below, I was persuaded that some variant of 41 had everything I wanted.

However, today I realize something that may or may not be important for my own music.

While 41Miracle Stud Loco may have 19 13/12's, etc. it doesn't have 13/12 *in relation to 1/1*. And the same all the way down the list.

So, there's always going to be a trade-off between consistent fingering, number of keys, and accuracy.

This is not hard to understand, but I keep hoping for a new solution.

Many times, I've read that such-and-such an EDO is a complete x-limit system. For instance, iirc, 58 EDO.

Today I'm going to see if somehow I can take a 44-note scale, and keep the epimorphism aspect so that all approximate 5ths (3/2s) are the same number of keys, same with all 4ths (4/3s) . So in Mod 44, it makes somewhat consistent patterns.

Then I'm going to design a two-octave version of the scale where the next-to-top note is removed to give a 4/1 on the very highest key.

The question that I don't know about is how much I can make the 3'2 in mod 44 not be too sharp before the result is ridiculous. That's the beauty of not having done this before.

I saw George Secor's In Defense of Inequality, and I've been a fan of Gene Ward Smith's epimorphic scales.

Perhaps there's something in 44 that is epimorphic and 4ths-5ths consistent or nearly so, and I only have to mutilate it slightly.

One of the advantages of being a relative newbie/ignoramous is that you don't quite know when something is impossible.

So you can tinker for *just one more day*...

hmm, here's 44, the 5th would sit at 709.091 and the 9th at 218.18, which is way too high:

0., 27.273, 54.545, 81.818, 109.091, 136.364, 163.636, 190.909, 218.182, 245.455, 272.727, 300., 327.273, 354.545, 381.818, 409.091, 436.364, 463.636, 490.909, 518.182, 545.455, 572.727, 600., 627.273, 654.545, 681.818, 709.091, 736.364, 763.636, 790.909, 818.182,

4ths want to be every 18 keys

5ths want to be every 26 keys

I only need a chain of good 5ths and 4ths moderately sharp, say 703.6, that lasts for 7 pitches in either direction.

That would mean that the following keyboard keys are spoken for:

chain of 4ths
k = 1 2 3 4 5 6 7
18*k= 18 36 10 28 2 20 38

chain of 5ths

k = 1 2 3 4 5 6 7
26*k= 26 8 34 16 42 24 6

Perhaps this chain could be extended a bit farther, as well, I don't know.

And everything else needs to fit between these, at least, if not more so.

It would be an ec-centric version of 22x2 notes, with fifths not so sharp, with one note lopped to allow for a 4/1 on top, on an 88-note controller.

Today's quixotic design adventure.

41 is probably better, though.

Hope springs eternal in the mind of the deluded one, but he recognizes reality, too.

-caleb

On Sep 15, 2010, at 8:54 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
> >
> > I wanted to make this "something missing" point a little more substantial.
> >
> > Specifically, in 41EDO I miss:
>
> Here is what is found in Miracle[41] ("Studloco")
>
> > 13/12 @ 138.57 cents
>
> 9 13/12s
>
> > 11/10 @ 165 cents
>
> 19 11/10s
>
> > 15/13 @ 247.7 cents
>
> 10 15/13s
>
> > 11/9 @ 347.4 cents
>
> 38 11/9s
>
> > and so forth.
>
> Vast amounts of and so forth.
>
> Below I give a particular tuning from the Scala directory, which could be tweaked to emphasize the intervals you most want.
>
> ! miracle41s.scl
> !
> Miracle-41 with Secor's minimax generator of 116.7155941 cents (5:9 exact). XH5, 1976
> 41
> !
> 32.84406
> 51.02748
> 83.87154
> 116.71559
> 149.55965
> 10/9
> 200.58713
> 233.43119
> 266.27525
> 299.11931
> 317.30272
> 350.14678
> 382.99084
> 415.83490
> 434.01832
> 466.86238
> 499.70643
> 532.55049
> 550.73391
> 583.57797
> 616.42203
> 649.26609
> 667.44951
> 700.29357
> 733.13762
> 765.98168
> 784.16510
> 817.00916
> 849.85322
> 882.69728
> 900.88069
> 933.72475
> 966.56881
> 999.41287
> 9/5
> 1050.44035
> 1083.28441
> 1116.12847
> 1134.31188
> 1167.15594
> 2/1
>
>

🔗caleb morgan <calebmrgn@...>

Here would be the framework that is consistent, before I start trying to see what I can fit.

I'm not smart enough to see in advance whether this has advantages

0 1/1
1
2 pyth b2-4ths chain, low!
3
4
5
6
7
8 9/8wide
9 8/7 goes here
10 32/27low
11
12
13
14 16/13 here
15 5/4!
16 81/64 high
17
18 4/3 low
19
20 4ths-chain tritone, low
21
22
23
24 5ths-chain tritone, high
25
26 3/2 high @ 703.6
27
28 128/81 low!
29 8/5!
30 13/8 here
31
32
33
34 27/16wider
35 7/4 goes here
36 16/9 low
37
38
39
40
41 15/8 is here!
42 pyth major 7th, higher than high!
43 (this note will be removed on top octave, to make top note be 4/1)
44

On Sep 17, 2010, at 9:57 AM, caleb morgan wrote:

>
> Just to clear up one thing.
>
> When I read the response below, I was persuaded that some variant of 41 had everything I wanted.
>
> However, today I realize something that may or may not be important for my own music.
>
> While 41Miracle Stud Loco may have 19 13/12's, etc. it doesn't have 13/12 *in relation to 1/1*. And the same all the way down the list.
>
> So, there's always going to be a trade-off between consistent fingering, number of keys, and accuracy.
>
> This is not hard to understand, but I keep hoping for a new solution.
>
> Many times, I've read that such-and-such an EDO is a complete x-limit system. For instance, iirc, 58 EDO.
>
> Today I'm going to see if somehow I can take a 44-note scale, and keep the epimorphism aspect so that all approximate 5ths (3/2s) are the same number of keys, same with all 4ths (4/3s) . So in Mod 44, it makes somewhat consistent patterns.
>
> Then I'm going to design a two-octave version of the scale where the next-to-top note is removed to give a 4/1 on the very highest key.
>
> The question that I don't know about is how much I can make the 3'2 in mod 44 not be too sharp before the result is ridiculous. That's the beauty of not having done this before.
>
> I saw George Secor's In Defense of Inequality, and I've been a fan of Gene Ward Smith's epimorphic scales.
>
> Perhaps there's something in 44 that is epimorphic and 4ths-5ths consistent or nearly so, and I only have to mutilate it slightly.
>
> One of the advantages of being a relative newbie/ignoramous is that you don't quite know when something is impossible.
>
> So you can tinker for *just one more day*...
>
> hmm, here's 44, the 5th would sit at 709.091 and the 9th at 218.18, which is way too high:
>
> 0., 27.273, 54.545, 81.818, 109.091, 136.364, 163.636, 190.909, 218.182, 245.455, 272.727, 300., 327.273, 354.545, 381.818, 409.091, 436.364, 463.636, 490.909, 518.182, 545.455, 572.727, 600., 627.273, 654.545, 681.818, 709.091, 736.364, 763.636, 790.909, 818.182,
>
> 4ths want to be every 18 keys
>
> 5ths want to be every 26 keys
>
> I only need a chain of good 5ths and 4ths moderately sharp, say 703.6, that lasts for 7 pitches in either direction.
>
> That would mean that the following keyboard keys are spoken for:
>
> chain of 4ths
> k = 1 2 3 4 5 6 7
> 18*k= 18 36 10 28 2 20 38
>
> chain of 5ths
>
> k = 1 2 3 4 5 6 7
> 26*k= 26 8 34 16 42 24 6
>
> Perhaps this chain could be extended a bit farther, as well, I don't know.
>
>
> And everything else needs to fit between these, at least, if not more so.
>
> It would be an ec-centric version of 22x2 notes, with fifths not so sharp, with one note lopped to allow for a 4/1 on top, on an 88-note controller.
>
> Today's quixotic design adventure.
>
>
> 41 is probably better, though.
>
> Hope springs eternal in the mind of the deluded one, but he recognizes reality, too.
>
>
>
> -caleb
>
>
>
>
>
> On Sep 15, 2010, at 8:54 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>> >
>> > I wanted to make this "something missing" point a little more substantial.
>> >
>> > Specifically, in 41EDO I miss:
>>
>> Here is what is found in Miracle[41] ("Studloco")
>>
>> > 13/12 @ 138.57 cents
>>
>> 9 13/12s
>>
>> > 11/10 @ 165 cents
>>
>> 19 11/10s
>>
>> > 15/13 @ 247.7 cents
>>
>> 10 15/13s
>>
>> > 11/9 @ 347.4 cents
>>
>> 38 11/9s
>>
>> > and so forth.
>>
>> Vast amounts of and so forth.
>>
>> Below I give a particular tuning from the Scala directory, which could be tweaked to emphasize the intervals you most want.
>>
>> ! miracle41s.scl
>> !
>> Miracle-41 with Secor's minimax generator of 116.7155941 cents (5:9 exact). XH5, 1976
>> 41
>> !
>> 32.84406
>> 51.02748
>> 83.87154
>> 116.71559
>> 149.55965
>> 10/9
>> 200.58713
>> 233.43119
>> 266.27525
>> 299.11931
>> 317.30272
>> 350.14678
>> 382.99084
>> 415.83490
>> 434.01832
>> 466.86238
>> 499.70643
>> 532.55049
>> 550.73391
>> 583.57797
>> 616.42203
>> 649.26609
>> 667.44951
>> 700.29357
>> 733.13762
>> 765.98168
>> 784.16510
>> 817.00916
>> 849.85322
>> 882.69728
>> 900.88069
>> 933.72475
>> 966.56881
>> 999.41287
>> 9/5
>> 1050.44035
>> 1083.28441
>> 1116.12847
>> 1134.31188
>> 1167.15594
>> 2/1
>>
>
>
>

🔗genewardsmith <genewardsmith@...>

🔗caleb morgan <calebmrgn@...>

🔗Carl Lumma <carl@...>

🔗caleb morgan <calebmrgn@...>

I've got a fever of 87.

I'm both using 87EDO as a tuning, but I'm also thinking in terms of an 87-key +1 (top octave) system.

By taking a scale of Gene Ward Smith that I had tweaked several times, and removing some notes so that it was down to 44,

I was able to come up with a scale that has all the pitches I want, plus operates in a 87-unit or 174-unit 2-octave tuning.

The top key on the keyboard is 4/1.

The pattern of intervals and srutis:

4, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 4, 4, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 4

0, 4, 6, 8, 10, 11, 13, 15, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 62, 64, 66, 68, 70, 72, 74, 76, 77, 79, 81, 83, 87, 91, 93, 95, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121, 123, 125, 127, 129, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 149, 151, 153, 155, 157, 159, 161, 163, 164, 166, 168, 170, 174

and the scale in cents:

0., 55.172, 82.759, 110.345, 137.931, 151.724, 179.31, 206.897, 234.483, 262.069, 289.655, 317.241, 344.828, 358.621, 386.207, 413.793, 441.379, 468.966, 496.552, 524.138, 551.724, 579.31, 593.103, 620.69, 648.276, 675.862, 703.448, 731.034, 758.621, 786.207, 813.793, 841.379, 855.172, 882.759, 910.345, 937.931, 965.517, 993.103, 1020.69, 1048.276, 1062.069, 1089.655, 1117.241, 1144.828, 1200., 1255.172, 1282.759, 1310.345, 1337.931, 1351.724, 1379.31, 1406.897, 1434.483, 1462.069, 1489.655, 1517.241, 1544.828, 1558.621, 1586.207, 1613.793, 1641.379, 1668.966, 1696.552, 1724.138, 1751.724, 1779.31, 1793.103, 1820.69, 1848.276, 1875.862, 1903.448, 1931.034, 1958.621, 1986.207, 2013.793, 2041.379, 2055.172, 2082.759, 2110.345, 2137.931, 2165.517, 2193.103, 2220.69, 2248.276, 2262.069, 2289.655, 2317.241, 2400.

and the intervals in cents:

55.172, 27.587, 27.586, 27.586, 13.793, 27.586, 27.587, 27.586, 27.586, 27.586, 27.586, 27.587, 13.793, 27.586, 27.586, 27.586, 27.587, 27.586, 27.586, 27.586, 27.586, 13.793, 27.587, 27.586, 27.586, 27.586, 27.586, 27.587, 27.586, 27.586, 27.586, 13.793, 27.587, 27.586, 27.586, 27.586, 27.586, 27.587, 27.586, 13.793, 27.586, 27.586, 27.587, 55.172, 55.172, 27.587, 27.586, 27.586, 13.793, 27.586, 27.587, 27.586, 27.586, 27.586, 27.586, 27.587, 13.793, 27.586, 27.586, 27.586, 27.587, 27.586, 27.586, 27.586, 27.586, 13.793, 27.587, 27.586, 27.586, 27.586, 27.586, 27.587, 27.586, 27.586, 27.586, 13.793, 27.587, 27.586, 27.586, 27.586, 27.586, 27.587, 27.586, 13.793, 27.586, 27.586, 82.759,

For some reason, thinking that 87 is 29x3, so this is a 400-cent symmetrical system, and seeing the pattern of srutis above, with the many stretches of 2's with a 1 every 7 or 8--this piques my combinatorial curiosity.

I bet there are many strange common-tone situations to be explored.

On Sep 17, 2010, at 12:40 PM, caleb morgan wrote:

>
> Heh.
>
> My obsession has led me to actually consider a non-standard keyboard.
>
> Are there any keyboards that are velocity-sensitive, have more than 120 or so keys, and are by companies that won't go out of business*, and don't cost over $1,000?
>
> [/Gripe mode] Why are there no companies that make standard keyboard with 97 keys (48x2 (+1)) [/Gripe mode]
>
>
> Otherwise, I might as well just settle down and learn 41 and variants.
>
> Its layout on a standard keyboard is actually sort of beautiful.
>
> And, I can use pitch-bend on the whole thing to move the 1/1 around with all the other pitches.
>
> -c
>
> *This is pretty important to me, as I've been burned before.
>
>
>
> On Sep 17, 2010, at 12:04 PM, genewardsmith wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>>
>> > While 41Miracle Stud Loco may have 19 13/12's, etc. it doesn't have 13/12 *in relation to 1/1*. And the same all the way down the list.
>>
>> You can move the 1/1 wherever you like.
>>
>
>
>

Compromise between 46EDO & Gene Ward Smith's 46 Epimorphic?

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