Yahoo Tuning Groups Ultimate Backup tuning So, I decided on using a 46-note framework. Got any scales w 46 notes?

All this searching, with so many dark nights of the soul, and so many moments of incompetence, has lead me to commit to a 46-note approach. What's weird about my scale, though, is that it's 46-pitch consistent up to the very last octave, where I have to chop out 4 pitches to fit into 88 keys.

This gives completely consistent fingering for almost everything up to the last 16/5 to 4/1, as it happens.

I already have a library of scales based on 46-note EDO, with 87EDO and JI approximations (nearest fits.)

Some are already copied here in my box:

http://www.box.net/shared/m37jhti1og#/shared/m37jhti1og/1/52696638

Then, I've made many versions that were conformed to various other high-number EDOs.

Eventually they'll all be backed up here, if anyone is interested.

These scales may be a little too idiosyncratic but they represent a satisfying solution, to me.

But I can't say, beyond putting them into piles with sharp fifths and flat ones, what the advantages of each variation are, by ear--except thinking some things sound nice, other things sound like problems. Very subtle, hard to hear, sometimes.

It's surprising how many things work.

But, perhaps someone might post some Scala files of good 46-note MOS scales they like?

Good 13-limit stuff?

Or 46-whatever scales...?

thanks,

caleb

🔗genewardsmith <genewardsmith@...>

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

> But, perhaps someone might post some Scala files of good 46-note MOS scales they like?
>
> Good 13-limit stuff?
>
> Or 46-whatever scales...?
>
> thanks,

A comma basis for 13-limit 46et is given by

[91/90, 121/120, 126/125, 169/168, 176/175]

This doesn't even come close to exhausting its superparticular commas, which also include 325/324, 351/350, 352/351, 364/363, 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655, 10648/10647 and 123201/123200. On top of that heap, you might toss on 847/845.

Taking any four independent 46et commas leads to a MOS scale, and these can also be used for Fokker blocks. Taking any three, two, one or none of them leads to a hobbit scale. This is a lot of scales. However, I will note that 126/125 with 176/175 gives thrush, and that a search could certainly be done for the lowest badness figures for other combinations of commas. The bottom line is, I'll have to get back to you on this one.

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
The bottom line is, I'll have to get back to you on this one.
>

Here is Thunor[46], a 46 note hobbit scale in one of the microtemperaments extending Thor, the 4375/4374 and 3025/3024 rank three temperament:

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

http://xenharmonic.wikispaces.com/Hobbits

! thunor46.scl
Thunor[46] hobbit in 494et
! Commas 4375/4374, 3025/3024, 1716/1715
46
!
31.57895
48.58300
80.16194
102.02429
133.60324
150.60729
182.18623
213.76518
235.62753
262.34818
284.21053
315.78947
332.79352
364.37247
395.95142
417.81377
449.39271
466.39676
497.97571
519.83806
546.55870
568.42105
600.00000
631.57895
648.58300
680.16194
702.02429
733.60324
750.60729
782.18623
813.76518
835.62753
862.34818
884.21053
915.78947
932.79352
964.37247
995.95142
1017.81377
1049.39271
1066.39676
1097.97571
1119.83806
1146.55870
1168.42105
1200.00000
!
! !prethunor46.scl
! Thunor[46] transversal
! 46
! !
! 891/875
! 250/243
! 22/21
! 297/280
! 27/25
! 275/252
! 10/9
! 198/175
! 8019/7000
! 220/189
! 33/28
! 6/5
! 1375/1134
! 100/81
! 44/35
! 891/700
! 162/125
! 55/42
! 4/3
! 27/20
! 1000/729
! 25/18
! 99/70
! 36/25
! 275/189
! 40/27
! 3/2
! 2673/1750
! 125/81
! 11/7
! 8/5
! 81/50
! 400/243
! 5/3
! 297/175
! 1250/729
! 110/63
! 16/9
! 9/5
! 8019/4375
! 50/27
! 66/35
! 2673/1400
! 1100/567
! 55/28
! 2/1

🔗Chris Vaisvil <chrisvaisvil@...>

I added it to the hobbit archive

http://micro.soonlabel.com/hobbit_scales/

On Tue, Oct 5, 2010 at 2:39 AM, genewardsmith
<genewardsmith@sbcglobal.net>wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "genewardsmith"
> <genewardsmith@...> wrote:
> The bottom line is, I'll have to get back to you on this one.
> >
>
> Here is Thunor[46], a 46 note hobbit scale in one of the microtemperaments
> extending Thor, the 4375/4374 and 3025/3024 rank three temperament:
>
> http://xenharmonic.wikispaces.com/Ragisma+family
>
> http://xenharmonic.wikispaces.com/Hobbits
>
> ! thunor46.scl
> Thunor[46] hobbit in 494et
> ! Commas 4375/4374, 3025/3024, 1716/1715
> 46
> !
> 31.57895
> 48.58300
> 80.16194
> 102.02429
> 133.60324
> 150.60729
> 182.18623
> 213.76518
> 235.62753
> 262.34818
> 284.21053
> 315.78947
> 332.79352
> 364.37247
> 395.95142
> 417.81377
> 449.39271
> 466.39676
> 497.97571
> 519.83806
> 546.55870
> 568.42105
> 600.00000
> 631.57895
> 648.58300
> 680.16194
> 702.02429
> 733.60324
> 750.60729
> 782.18623
> 813.76518
> 835.62753
> 862.34818
> 884.21053
> 915.78947
> 932.79352
> 964.37247
> 995.95142
> 1017.81377
> 1049.39271
> 1066.39676
> 1097.97571
> 1119.83806
> 1146.55870
> 1168.42105
> 1200.00000
> !
> ! !prethunor46.scl
> ! Thunor[46] transversal
> ! 46
> ! !
> ! 891/875
> ! 250/243
> ! 22/21
> ! 297/280
> ! 27/25
> ! 275/252
> ! 10/9
> ! 198/175
> ! 8019/7000
> ! 220/189
> ! 33/28
> ! 6/5
> ! 1375/1134
> ! 100/81
> ! 44/35
> ! 891/700
> ! 162/125
> ! 55/42
> ! 4/3
> ! 27/20
> ! 1000/729
> ! 25/18
> ! 99/70
> ! 36/25
> ! 275/189
> ! 40/27
> ! 3/2
> ! 2673/1750
> ! 125/81
> ! 11/7
> ! 8/5
> ! 81/50
> ! 400/243
> ! 5/3
> ! 297/175
> ! 1250/729
> ! 110/63
> ! 16/9
> ! 9/5
> ! 8019/4375
> ! 50/27
> ! 66/35
> ! 2673/1400
> ! 1100/567
> ! 55/28
> ! 2/1
>
>
>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> I added it to the hobbit archive
>
> http://micro.soonlabel.com/hobbit_scales/

Here's another 46 note 13-limit hobbit.

! akea46_13.scl
Tridecimal Akea[46] hobbit minimax tuning
! Commas 325/324, 352/351, 385/384
46
!
26.52580
53.05160
67.35084
111.95263
138.47843
152.77767
179.30347
205.82927
232.35507
264.73030
291.25610
317.78190
344.30770
358.60694
385.13274
411.65854
438.18434
470.55956
497.08537
523.61117
550.13697
564.43621
590.96201
635.56379
649.86303
676.38883
702.91463
729.44044
755.96624
788.34146
814.86726
841.39306
855.69230
882.21810
908.74390
935.26970
967.64493
994.17073
1020.69653
1047.22233
1061.52157
1088.04737
1114.57317
1146.94840
1173.47420
1200.00000
!
! ! preakea46.scl
! Akea[46] transversal
! 46
! !
! 55/54
! 33/32
! 25/24
! 16/15
! 13/12
! 12/11
! 10/9
! 9/8
! 8/7
! 7/6
! 13/11
! 6/5
! 11/9
! 16/13
! 5/4
! 80/63
! 9/7
! 21/16
! 4/3
! 27/20
! 11/8
! 18/13
! 45/32
! 13/9
! 16/11
! 40/27
! 3/2
! 32/21
! 54/35
! 52/33
! 8/5
! 13/8
! 18/11
! 5/3
! 22/13
! 12/7
! 7/4
! 16/9
! 9/5
! 11/6
! 24/13
! 15/8
! 40/21
! 35/18
! 65/33
! 2/1

🔗Chris Vaisvil <chrisvaisvil@...>

🔗caleb morgan <calebmrgn@...>

Awesome, keep em' comin'.

Thanks.

I also like, from a while ago, Rodan46.

Here's how I fit 46x2 into 88 keys:

! Rodan 46 (GB)
87-n 234.48 generator ca. 28&14cent steps
87
!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
1227.586,
1255.172,
1282.758,
1310.344,
1324.137,
1351.723,
1379.309,
1406.895,
1434.481,
1462.066,
1489.652,
1517.238,
1544.824,
1558.617,
1586.203,
1613.789,
1641.375,
1668.961,
1696.547,
1724.133,
1751.719,
1779.305,
1793.098,
1820.684,
1848.27,
1875.856,
1903.442,
1986.199,
2013.785,
2041.371,
2055.164,
2082.75,
2110.336,
2165.508,
2193.094,
2220.68,
2248.266,
2275.852,
2289.645,
2317.231,
2400.0

On Oct 5, 2010, at 6:04 PM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > I added it to the hobbit archive
> >
> > http://micro.soonlabel.com/hobbit_scales/
>
> Here's another 46 note 13-limit hobbit.
>
> ! akea46_13.scl
> Tridecimal Akea[46] hobbit minimax tuning
> ! Commas 325/324, 352/351, 385/384
> 46
> !
> 26.52580
> 53.05160
> 67.35084
> 111.95263
> 138.47843
> 152.77767
> 179.30347
> 205.82927
> 232.35507
> 264.73030
> 291.25610
> 317.78190
> 344.30770
> 358.60694
> 385.13274
> 411.65854
> 438.18434
> 470.55956
> 497.08537
> 523.61117
> 550.13697
> 564.43621
> 590.96201
> 635.56379
> 649.86303
> 676.38883
> 702.91463
> 729.44044
> 755.96624
> 788.34146
> 814.86726
> 841.39306
> 855.69230
> 882.21810
> 908.74390
> 935.26970
> 967.64493
> 994.17073
> 1020.69653
> 1047.22233
> 1061.52157
> 1088.04737
> 1114.57317
> 1146.94840
> 1173.47420
> 1200.00000
> !
> ! ! preakea46.scl
> ! Akea[46] transversal
> ! 46
> ! !
> ! 55/54
> ! 33/32
> ! 25/24
> ! 16/15
> ! 13/12
> ! 12/11
> ! 10/9
> ! 9/8
> ! 8/7
> ! 7/6
> ! 13/11
> ! 6/5
> ! 11/9
> ! 16/13
> ! 5/4
> ! 80/63
> ! 9/7
> ! 21/16
> ! 4/3
> ! 27/20
> ! 11/8
> ! 18/13
> ! 45/32
> ! 13/9
> ! 16/11
> ! 40/27
> ! 3/2
> ! 32/21
> ! 54/35
> ! 52/33
> ! 8/5
> ! 13/8
> ! 18/11
> ! 5/3
> ! 22/13
> ! 12/7
> ! 7/4
> ! 16/9
> ! 9/5
> ! 11/6
> ! 24/13
> ! 15/8
> ! 40/21
> ! 35/18
> ! 65/33
> ! 2/1
>
>

🔗caleb morgan <calebmrgn@...>

Thank you for this. A previous post of mine seems to have gotten lost. I said I liked the 46-pitch version of Rodan, and posted it.

I'd be interested in more things like Rodan for 46 notes.

Here's a close-to-46 JI-ish scale for all 88 keys, basic design by Gene W. S. with tweaks by me.

! caleb87.scl
87 note 13-lim tweaked epi by G.W.Smith mod3cm
87
! 8,7,6,6,5,5,5,4,
30.0
53.0
21/20
15/14
13/12
150.68
179.1
207.2
8/7
7/6
13/11
313.6
11/9
16/13
385.0
14/11
9/7
21/16
496.4
27/20
11/8
7/5
593.0
!
10/7
648.68
40/27
703.6
32/21
14/9
11/7
819.0
13/8
18/11
882.7
22/13
12/7
7/4
992.8
1013.6
11/6
24/13
1085.0
40/21
1147
1170.0
2/1
1230.0
1253.0
21/10
15/7
13/6
1350.68
1379.1
1407.2
16/7
7/3 !
26/11
1513.6
22/9
32/13
1585.0
28/11
18/7
21/8
1696.4
27/10
11/4
14/5
1793.0
20/7 !a
1848.68
80/27 !b
1903.6
22/7
2019.0
26/8
36/11
2082.7
44/13
14/4
2192.8
2213.6
22/6
48/13
2285.0
80/21
4/1

On Oct 5, 2010, at 2:39 AM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> The bottom line is, I'll have to get back to you on this one.
> >
>
> Here is Thunor[46], a 46 note hobbit scale in one of the microtemperaments extending Thor, the 4375/4374 and 3025/3024 rank three temperament:
>
> http://xenharmonic.wikispaces.com/Ragisma+family
>
> http://xenharmonic.wikispaces.com/Hobbits
>
> ! thunor46.scl
> Thunor[46] hobbit in 494et
> ! Commas 4375/4374, 3025/3024, 1716/1715
> 46
> !
> 31.57895
> 48.58300
> 80.16194
> 102.02429
> 133.60324
> 150.60729
> 182.18623
> 213.76518
> 235.62753
> 262.34818
> 284.21053
> 315.78947
> 332.79352
> 364.37247
> 395.95142
> 417.81377
> 449.39271
> 466.39676
> 497.97571
> 519.83806
> 546.55870
> 568.42105
> 600.00000
> 631.57895
> 648.58300
> 680.16194
> 702.02429
> 733.60324
> 750.60729
> 782.18623
> 813.76518
> 835.62753
> 862.34818
> 884.21053
> 915.78947
> 932.79352
> 964.37247
> 995.95142
> 1017.81377
> 1049.39271
> 1066.39676
> 1097.97571
> 1119.83806
> 1146.55870
> 1168.42105
> 1200.00000
> !
> ! !prethunor46.scl
> ! Thunor[46] transversal
> ! 46
> ! !
> ! 891/875
> ! 250/243
> ! 22/21
> ! 297/280
> ! 27/25
> ! 275/252
> ! 10/9
> ! 198/175
> ! 8019/7000
> ! 220/189
> ! 33/28
> ! 6/5
> ! 1375/1134
> ! 100/81
> ! 44/35
> ! 891/700
> ! 162/125
> ! 55/42
> ! 4/3
> ! 27/20
> ! 1000/729
> ! 25/18
> ! 99/70
> ! 36/25
> ! 275/189
> ! 40/27
> ! 3/2
> ! 2673/1750
> ! 125/81
> ! 11/7
> ! 8/5
> ! 81/50
> ! 400/243
> ! 5/3
> ! 297/175
> ! 1250/729
> ! 110/63
> ! 16/9
> ! 9/5
> ! 8019/4375
> ! 50/27
> ! 66/35
> ! 2673/1400
> ! 1100/567
> ! 55/28
> ! 2/1
>
>

🔗Graham Breed <gbreed@...>

On 7 October 2010 16:21, caleb morgan <calebmrgn@...> wrote:

>
>
> Thank you for this. A previous post of mine seems to have gotten lost. I
> said I liked the 46-pitch version of Rodan, and posted it.
>

I saw that!

> I'd be interested in more things like Rodan for 46 notes.
>

Here are some things with 46 notes to a pure octave. One of them is Rodan.
See if it looks familiar.

Unidec
18.427
50.406
68.833
100.812
119.239
151.219
169.646
201.625
233.604
252.031
284.010
302.437
334.417
352.844
384.823
416.802
435.229
467.208
485.635
517.615
536.042
568.021
600.000
618.427
650.406
668.833
700.812
719.239
751.219
769.646
801.625
833.604
852.031
884.010
902.437
934.417
952.844
984.823
1016.802
1035.229
1067.208
1085.635
1117.615
1136.042
1168.021
1200.000

Rodan
13.768
41.357
68.946
96.536
124.125
151.714
179.304
206.893
234.482
248.250
275.839
303.428
331.018
358.607
386.196
413.786
441.375
468.964
482.732
510.321
537.911
565.500
593.089
620.678
648.268
675.857
703.446
717.214
744.803
772.393
799.982
827.571
855.161
882.750
910.339
937.929
951.696
979.286
1006.875
1034.464
1062.053
1089.643
1117.232
1144.821
1172.411
1200.000

Diaschismic
22.222
44.444
66.666
103.704
125.926
148.148
170.370
207.407
229.629
251.851
274.073
311.111
333.333
355.555
377.777
414.815
437.037
459.259
481.481
518.518
540.740
562.962
600.000
622.222
644.444
666.666
703.704
725.926
748.148
770.370
807.407
829.629
851.851
874.073
911.111
933.333
955.555
977.777
1014.815
1037.037
1059.259
1081.481
1118.518
1140.740
1162.962
1200.000

Valentino
16.698
47.328
77.958
94.656
125.286
155.916
172.614
203.244
233.874
250.572
281.202
311.832
328.530
359.160
389.790
406.488
437.118
467.748
484.446
515.076
545.706
562.404
593.034
623.664
640.362
670.992
701.622
718.320
748.950
779.580
796.278
826.908
857.538
874.236
904.866
935.496
952.194
982.824
1013.454
1030.152
1060.782
1091.412
1108.110
1138.740
1169.370
1200.000

🔗caleb morgan <calebmrgn@...>

🔗Graham Breed <gbreed@...>

🔗caleb morgan <calebmrgn@...>

can someone make a Scala file out of this and explain how/why they did it?

Portent 2 dimensions higher

Equal Temperament Mappings
2 3 5 7 11 13 17
[< 31 49 72 87 107 115 127 ]
< 72 114 167 202 249 266 294 ]
< 46 73 107 129 159 170 188 ]>

Reduced Mapping
2 3 5 7 11 13 17
[< 1 1 0 3 5 1 1 ]
< 0 3 0 -1 4 -10 -8 ]
< 0 0 1 0 -1 2 2 ]>

Generator Tunings (cents)
[1200.675, 233.563, 2786.856>

Step Tunings (cents)
[5.671, 11.296, 4.600>

Tuning Map (cents)
<1200.675, 1901.363, 2786.856, 3368.463, 4150.771, 4438.762, 4905.887]

Complexity 0.269954
Adjusted Error 1.522717 cents
TOP-RMS Error 0.372533 cents/octave

temperament finding scripts

On Oct 13, 2010, at 10:50 AM, caleb morgan wrote:

> some of the 46-note tunings are here, Scala format. Three pages.
>
>
> http://www.box.net/shared/m37jhti1og#/shared/m37jhti1og/1/52696638
>
> If people want to make up and send 46-pitch tunings, I'd love to add them to my collection.
>
> For daily use, I mostly use 46, JI with 46, and 46 subset of 87.
>
> I'd love to add to the "small fifth" side of my collection.
>
> caleb
>
>

🔗Graham Breed <gbreed@...>

🔗caleb morgan <calebmrgn@...>

Thanks, that looks like an interesting scale to me!
Playing it now.

caleb

On Oct 14, 2010, at 11:43 AM, Graham Breed wrote:

> caleb morgan <calebmrgn@...> wrote:
> >
> > can someone make a Scala file out of this and explain
> > how/why they did it?
>
> What I did is: go to my Python interpreter, recreate the
> temperament (it uses the best mappings of 31, 46, and 72)
> and use the method I'd already written, which involves
> rotations of maximally even scales, to produce a 46 note
> scale.
>
> Then I checked it, and wondered why it had four
> distinct step sizes, when it should be possible to reduce
> this to two. And also why the smallest step is less than
> half the size of the biggest step. But I haven't been able
> to come up with anything better, so here it is, and we'll
> see how everybody else does.
>
> 15.895
> 48.757
> 70.324
> 97.515
> 119.081
> 151.943
> 167.839
> 200.701
> 233.563
> 249.458
> 282.320
> 303.886
> 331.078
> 352.644
> 385.506
> 412.697
> 434.263
> 467.125
> 483.021
> 515.883
> 537.449
> 564.640
> 597.502
> 619.068
> 646.260
> 667.826
> 700.688
> 716.583
> 749.445
> 771.012
> 798.203
> 831.065
> 852.631
> 879.822
> 901.389
> 934.251
> 950.146
> 983.008
> 1015.870
> 1031.765
> 1064.628
> 1086.194
> 1113.385
> 1134.951
> 1167.813
> 1200.675
>
>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@> wrote:
> >
> >
> > can someone make a Scala file out of this and explain how/why they did it?
>
> I could compute the hobbit scale for it.

Here is it:

! portent46.scl
!
17-limit Portent[46] hobbit
46
!
32.85839
48.26285
81.12123
104.04424
129.38408
152.30709
185.16548
200.56994
233.42832
266.28671
281.69117
314.54956
347.40794
362.81240
385.73541
418.59380
451.45218
466.85665
499.71503
515.11949
547.97788
580.83626
603.75927
619.16374
652.02212
684.88051
700.28497
733.14335
766.00174
781.40620
814.26459
837.18760
862.52743
885.45044
918.30883
933.71329
966.57168
999.43006
1014.83452
1047.69291
1070.61592
1095.95576
1118.87877
1151.73715
1167.14161
1200.00000
!
! ! preportent46.scl
! 17-limit transversal
! 17
! !
! 49/48
! 36/35
! 21/20
! 17/16
! 14/13
! 12/11
! 1715/1536
! 384/343
! 8/7
! 7/6
! 153/130
! 6/5
! 17/14
! 16/13
! 96/77
! 245/192
! 3072/2401
! 64/49
! 343/256
! 27/20
! 48/35
! 7/5
! 24/17
! 10/7
! 35/24
! 40/27
! 512/343
! 49/32
! 20/13
! 384/245
! 77/48
! 13/8
! 18/11
! 5/3
! 245/144
! 12/7
! 7/4
! 343/192
! 3072/1715
! 119/65
! 13/7
! 32/17
! 21/11
! 35/18
! 96/49
! 2/1

🔗Graham Breed <gbreed@...>

caleb morgan <calebmrgn@...> wrote:
>
> can someone make a Scala file out of this and explain
> how/why they did it?

I now have the equivalent of an MOS for this. It wasn't an
easy process, and dealing with the full generality is for
tuning-math. But I'll explain what I did here as best as I
can.

> Equal Temperament Mappings
> 2 3 5 7 11 13 17
> [< 31 49 72 87 107 115
> 127 ] < 72 114 167 202
> 249 266 294 ] < 46 73 107
> 129 159 170 188 ]>

My e-mail client messed up the word-wrapping of this, so I
can't be sure I reconstructed it right. The information we
need from it, though, is that that Portent involves 31, 72,
and 46 note scales.

> Step Tunings (cents)
> [5.671, 11.296, 4.600>

This tells us that an octave is made up of 31 steps of
5.671 cents, 72 steps of 11.296 cents, and 46 steps of
4.600 cents.

Check:

>>> 5.671*31 + 11.296*72 + 46*4.600
1200.713

That's a bit higher than the tuning map, but about right
given the accumulated rounding error:

> Tuning Map (cents)
> <1200.675, 1901.363, 2786.856, 3368.463, 4150.771,
> 4438.762, 4905.887]

That's why I'll keep the step sizes as three figure
decimals in cents, despite there being complaints in the
past about this being excessive.

So, that defines the tuning, but we end up with too many
notes. We don't want 72 of anything, because 72>46. So
replace 72 with 72-46=26. That leaves us with 26, 31, and
46 notes. 26+31=57, so you can't describe a 46 note with
these steps. Replace 31 with 46-31=15, and we have 15, 26,
and 46 notes. This is enough to describe a 46 note scale
with steps of one size, 26 of another, and 46-15-26=5 steps
of another.

We now have the portentous numbers 5, 15, and 26. I don't
know if they uniquely solve the problem of a 46 note
portent. In the rank 2 case they are unique. A 12 note
meantone is always 7 of one step size and 5 of another, for
example.

I happen to have a function that can give the correct
mappings for 5, 15, and 26 note equal temperaments
consistent with the portent mapping. They are:

[<26, 41, 60, 73, 90, 96, 106],
<15, 24, 35, 42, 52, 55, 61],
<5, 8, 12, 14, 17, 19, 21]>

Plugging that rank 3 mapping back into the temperament
constructor gives step sizes of 32.862, 15.895, 21.566
cents.

Check:

>>> 26*32.862 + 15*15.895 + 5*21.566
1200.6669999999999

You could also have calculated these step sizes (plus
rounding error) by combining the original step sizes.

Now I need to work out the best distribution of these three
step sizes. I'll ignore the 26 large steps and look at 15
of one size and 5 of t'other in the 46 note octave.

A 46 from 15 note maximally even scale is:

0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1

This shows the 46 note scale in intervals of the 15 note
scale. The 1 tells us "use the 15.895 cent step" and the
0 tells us "use another step".

Here's a 46 from 5 note maximally even scale:

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

The 1 tells us "use the 21.566 note step" and the 0 tells
us "use another step". Where both scales have a zero, we
use the 32.862 cent step.

There's a problem, though. Sometimes both scales have a 1
in the same place. To get a scale with the right step
sizes, that shouldn't happen. So we need a different
rotation of the 5 from 46 scale.

There's another constraint I'll apply. The 15 from 46
scale starts with three zeros. That translates to three
consecutive steps of the same size. You don't get this
anywhere else in the scale. For maximal variety, I'll
choose a rotation of the 5 from 46 scale so that the middle
step of those three becomes the 32.862 cent step.

There are 5 rotations that work with both these constraints.
I happened to choose this one:

0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

Combining it with the 15 from 46 scale above gives this key
to the 46 note scale:

0 2 0 1 0 0 1 0 0 1 2 0 1 0 0 1 0 0 1 2 0 1 0 0 1 0 0 1 2 0
1 0 0 1 0 0 1 2 0 1 0 0 1 0 0 1

Where there's a 0 it means use the 32.862 cent step, where
there's a 1 it means use the 15.896 cent step, and where
there's a 2 it means use the 21.566 cent step.

Here is that scale:

32.862
54.428
87.290
103.186
136.048
168.910
184.805
217.667
250.529
266.425
287.991
320.853
336.748
369.610
402.473
418.368
451.230
484.092
499.987
521.554
554.416
570.311
603.173
636.035
651.931
684.793
717.655
733.550
755.116
787.978
803.874
836.736
869.598
885.493
918.355
951.217
967.113
988.679
1021.541
1037.436
1070.298
1103.160
1119.056
1151.918
1184.780
1200.675

Graham

🔗Chris Vaisvil <chrisvaisvil@...>

🔗caleb morgan <calebmrgn@...>

Thank you, what an amazing process.

Describing it fully like this helps me a lot, thanks for the effort.

The tuning is in my collection, I'm checking it out.

One way to amass a collection of scales is to tweak scales by Gene Ward Smith.

I wonder what possible flavors I'm missing.

How can I have a "complete" collection--or is the notion even coherent?

46 is a lot of pitches--It ought to be able to have just about *any* flavor.

What should I add to my collection?

Thank you both for these brilliant scales.

And, Mr. Smith, thanks for including the precursor JI versions. They are great to have.

Caleb

On Oct 15, 2010, at 6:08 AM, Graham Breed wrote:

> caleb morgan <calebmrgn@...> wrote:
> >
> > can someone make a Scala file out of this and explain
> > how/why they did it?
>
> I now have the equivalent of an MOS for this. It wasn't an
> easy process, and dealing with the full generality is for
> tuning-math. But I'll explain what I did here as best as I
> can.
>
> > Equal Temperament Mappings
> > 2 3 5 7 11 13 17
> > [< 31 49 72 87 107 115
> > 127 ] < 72 114 167 202
> > 249 266 294 ] < 46 73 107
> > 129 159 170 188 ]>
>
> My e-mail client messed up the word-wrapping of this, so I
> can't be sure I reconstructed it right. The information we
> need from it, though, is that that Portent involves 31, 72,
> and 46 note scales.
>
> > Step Tunings (cents)
> > [5.671, 11.296, 4.600>
>
> This tells us that an octave is made up of 31 steps of
> 5.671 cents, 72 steps of 11.296 cents, and 46 steps of
> 4.600 cents.
>
> Check:
>
> >>> 5.671*31 + 11.296*72 + 46*4.600
> 1200.713
>
> That's a bit higher than the tuning map, but about right
> given the accumulated rounding error:
>
> > Tuning Map (cents)
> > <1200.675, 1901.363, 2786.856, 3368.463, 4150.771,
> > 4438.762, 4905.887]
>
> That's why I'll keep the step sizes as three figure
> decimals in cents, despite there being complaints in the
> past about this being excessive.
>
> So, that defines the tuning, but we end up with too many
> notes. We don't want 72 of anything, because 72>46. So
> replace 72 with 72-46=26. That leaves us with 26, 31, and
> 46 notes. 26+31=57, so you can't describe a 46 note with
> these steps. Replace 31 with 46-31=15, and we have 15, 26,
> and 46 notes. This is enough to describe a 46 note scale
> with steps of one size, 26 of another, and 46-15-26=5 steps
> of another.
>
> We now have the portentous numbers 5, 15, and 26. I don't
> know if they uniquely solve the problem of a 46 note
> portent. In the rank 2 case they are unique. A 12 note
> meantone is always 7 of one step size and 5 of another, for
> example.
>
> I happen to have a function that can give the correct
> mappings for 5, 15, and 26 note equal temperaments
> consistent with the portent mapping. They are:
>
> [<26, 41, 60, 73, 90, 96, 106],
> <15, 24, 35, 42, 52, 55, 61],
> <5, 8, 12, 14, 17, 19, 21]>
>
> Plugging that rank 3 mapping back into the temperament
> constructor gives step sizes of 32.862, 15.895, 21.566
> cents.
>
> Check:
>
> >>> 26*32.862 + 15*15.895 + 5*21.566
> 1200.6669999999999
>
> You could also have calculated these step sizes (plus
> rounding error) by combining the original step sizes.
>
> Now I need to work out the best distribution of these three
> step sizes. I'll ignore the 26 large steps and look at 15
> of one size and 5 of t'other in the 46 note octave.
>
> A 46 from 15 note maximally even scale is:
>
> 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0
> 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1
>
> This shows the 46 note scale in intervals of the 15 note
> scale. The 1 tells us "use the 15.895 cent step" and the
> 0 tells us "use another step".
>
> Here's a 46 from 5 note maximally even scale:
>
> 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
> 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
>
> The 1 tells us "use the 21.566 note step" and the 0 tells
> us "use another step". Where both scales have a zero, we
> use the 32.862 cent step.
>
> There's a problem, though. Sometimes both scales have a 1
> in the same place. To get a scale with the right step
> sizes, that shouldn't happen. So we need a different
> rotation of the 5 from 46 scale.
>
> There's another constraint I'll apply. The 15 from 46
> scale starts with three zeros. That translates to three
> consecutive steps of the same size. You don't get this
> anywhere else in the scale. For maximal variety, I'll
> choose a rotation of the 5 from 46 scale so that the middle
> step of those three becomes the 32.862 cent step.
>
> There are 5 rotations that work with both these constraints.
> I happened to choose this one:
>
> 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
> 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
>
> Combining it with the 15 from 46 scale above gives this key
> to the 46 note scale:
>
> 0 2 0 1 0 0 1 0 0 1 2 0 1 0 0 1 0 0 1 2 0 1 0 0 1 0 0 1 2 0
> 1 0 0 1 0 0 1 2 0 1 0 0 1 0 0 1
>
> Where there's a 0 it means use the 32.862 cent step, where
> there's a 1 it means use the 15.896 cent step, and where
> there's a 2 it means use the 21.566 cent step.
>
> Here is that scale:
>
> 32.862
> 54.428
> 87.290
> 103.186
> 136.048
> 168.910
> 184.805
> 217.667
> 250.529
> 266.425
> 287.991
> 320.853
> 336.748
> 369.610
> 402.473
> 418.368
> 451.230
> 484.092
> 499.987
> 521.554
> 554.416
> 570.311
> 603.173
> 636.035
> 651.931
> 684.793
> 717.655
> 733.550
> 755.116
> 787.978
> 803.874
> 836.736
> 869.598
> 885.493
> 918.355
> 951.217
> 967.113
> 988.679
> 1021.541
> 1037.436
> 1070.298
> 1103.160
> 1119.056
> 1151.918
> 1184.780
> 1200.675
>
> Graham
>

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

🔗caleb morgan <calebmrgn@...>

> ...
> What should I add to my collection?
>
>
> Caleb
>
>
>
>
>
> ._,_.___
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http://x31eq.com/catalog.htm

I'm going to come up with some examples of 46-note versions of each of these linear temperaments as a way to having a complete set of scales.

Slightly bizarre is my first effort, where I simply combine 19 and 31 tones with a few 46 thrown in for this 46-note scale:

!46-n 19&31&46 bizarro
19&31 bizarro with a little 46
46
!
38.71
63.158
77.419
116.129
126.316
154.839
189.474
193.548
232.258
252.632
270.968
!
315.789
348.387
365.2174
378.947
387.097
442.105
464.516
505.263
521.7391
541.935
568.421
600.00
!
620.689
631.579
658.065
694.737
735.484
757.895
774.194
812.903
821.053
851.613
884.211
890.323
!
929.032
947.368
967.742
1010.526
1045.161
1073.684
1083.871
1122.581
1136.842
1161.29
1200.0

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗dasdastri <dasdastri@...>

Not that it really matters much, as it doesn't at all, but the way I've seen this for the past 25 years-or-so is by scaling things in the direction of palindromic symmetry--in other words rounding both up and down depending on the period's mean.
I've always contended that periodicity was not octave specific on this list BTW, even when it was not a particularly popular view and that palindromic symmetry was maximal evenness in the way it was embedded by in my mind by others who explained Mayhill et al.to me second-hand.
I know it's wise to seek the source before opening the mouth, but I was (for better or worse) pretty sure on/convinced of this one from day one.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > On 15 October 2010 23:10, genewardsmith <genewardsmith@> wrote:
>
> > That looks like distributional evenness. I got the maximal evenness
> > formula from an Agmon paper, and it's proven consistent with the
> > original definition in a paper I don't have. For an n note scale
> > taken from d note equal temperament, you take n equal divisions and
> > round them either up or down or somehow consistently to fit the d
> > steps. I go for down, meaning the "floor" function, or truncation of
> > positive numbers.
>
> In other words, a maximally even scale is the closest approximation in n edo to m edo where m < n. What we have, I presume, is that sum of floor functions construction which generalizes MOS and Fokker blocks and which we've discussed on tuning-math.
>

So, I decided on using a 46-note framework. Got any scales w 46 notes?

🔗calebmrgn <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗caleb morgan <calebmrgn@...>

🔗caleb morgan <calebmrgn@...>

🔗Graham Breed <gbreed@...>

🔗caleb morgan <calebmrgn@...>

🔗Graham Breed <gbreed@...>

🔗caleb morgan <calebmrgn@...>

🔗caleb morgan <calebmrgn@...>

🔗Graham Breed <gbreed@...>

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗caleb morgan <calebmrgn@...>

🔗caleb morgan <calebmrgn@...>

🔗caleb morgan <calebmrgn@...>

🔗genewardsmith <genewardsmith@...>

🔗Graham Breed <gbreed@...>

🔗caleb morgan <calebmrgn@...>

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗dasdastri <dasdastri@...>