Tuning-Math Digests messages 9075 - 9099

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 10

Previous Next

9000 9050 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 9600 9650 9700 9750 9800 9850 9900 9950

9050 - 9075 -



top of page bottom of page down


Message: 9075

Date: Fri, 09 Jan 2004 21:16:36

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Carl Lumma

>>So Graham had the all the digits right, I just needed more precision.
>>Multiply by 12, and we get
>>1197.67406985219          1896.31727726597          2794.57282965511
>>Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.
>>
>>
>>           10            9       199.61       17.209       2.6508
>>            9            8       199.61       4.2977      0.69655
>>            6            5       299.42       16.223       3.3061
>>            5            4       399.22       12.911       2.9873
>>            4            3       499.03      0.98586        0.275
>>            3            2       698.64       3.3118       1.2812
>>            8            5       798.45       15.237        2.863
>>            5            3       898.26       13.897        3.557
>>            9            5       998.06       19.535        3.557
>>            2            1       1197.7       2.3259       2.3259
>>            9            4       1397.3       6.6236       1.2812
>>           12            5       1497.1       18.549       3.1402
>>            5            2       1596.9       10.585       3.1864
>>            8            3       1696.7       1.3401      0.29227
>>            3            1       1896.3       5.6377        3.557
>>           16            5       1996.1       17.563       2.7781
>>           10            3       2095.9       11.571       2.3581
>>           18            5       2195.7        21.86       3.3674
>>           15            4       2295.5       7.2733       1.2313
>>            4            1       2395.3       4.6519       2.3259
>>            9            2         2595       8.9495       2.1462
>>            5            1       2794.6       8.2591        3.557
>>           16            3       2894.4        3.666       0.6564
>>            6            1         3094       7.9637       3.0808
>>           25            4       3193.8        21.17       3.1864
>>           20            3       3293.6        9.245       1.5651
>>           15            2       3493.2       4.9473       1.0082
>>            8            1         3593       6.9778       2.3259
>>           25            3       3692.8       22.156        3.557
>>            9            1       3792.6       11.275        3.557
>>           10            1       3992.2       5.9332       1.7861
>>           32            3       4092.1       5.9919      0.90994
>>           12            1       4291.7        10.29       2.8702
>>           25            2       4391.5       18.844       3.3389
>>           27            2       4491.3       14.587       2.5348
>>           15            1       4690.9       2.6214      0.67097
>>           16            1       4790.7       9.3037       2.3259
>>           18            1       4990.3       13.601       3.2618
>>           20            1       5189.9       3.6073      0.83464
>>           45            2       5389.5       0.6904      0.10635
>>           24            1       5489.3       12.616       2.7515
>>           25            1       5589.1       16.518        3.557
>>           27            1         5689       16.913        3.557
>>           30            1       5888.6      0.29546     0.060214
>>           32            1       5988.4        11.63       2.3259
>>           36            1         6188       15.927       3.0808
>>           40            1       6387.6       1.2813      0.24076
>>           45            1       6587.2       3.0163      0.54924
>>           48            1         6687       14.941       2.6753
>>           50            1       6786.8       14.192       2.5146
>>           54            1       6886.6       19.239       3.3431
>>           60            1       7086.2       2.0305      0.34375
>>           64            1         7186       13.956       2.3259
>>           72            1       7385.7       18.253       2.9584
>>           75            1       7485.5       10.881       1.7468
>>           80            1       7585.3       1.0446      0.16524
>>           81            1       7585.3       22.551        3.557
>>           90            1       7784.9       5.3423      0.82292
>>           96            1       7884.7       17.267       2.6222
>>          100            1       7984.5       11.866       1.7861
>
>The alaska tunings are essentially circulating versions of this
>tuning.

Which was based on...

! zeta12.scl
!
12 equal zeta tuning
 12
!
99.807
199.614
299.422
399.229
499.036
598.843
698.650
798.457
898.265
998.072
1097.879
1197.686

...Notice the similarity...

-Carl


top of page bottom of page up down


Message: 9076

Date: Fri, 09 Jan 2004 08:55:36

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >I don't know, but I plan on investigating TOP tunings of equal and 
> >planar temperaments as well as linear ones. Presumably one gets a 
> >squashing. The octaves of Dom7 are pretty short.
> 
> You have a way of combining commas, then?
> 
> -Carl

Seemingly -- Pajara uses two commas, after all.

I'm trying to reproduce Gene's results, but I probably need to think 
about it away from the computer . . .


top of page bottom of page up down


Message: 9077

Date: Fri, 09 Jan 2004 23:57:30

Subject: also...

From: Carl Lumma

>>>           10            9       199.61       17.209       2.6508
>>>            9            8       199.61       4.2977      0.69655
>>>            6            5       299.42       16.223       3.3061
>>>            5            4       399.22       12.911       2.9873
>>>            4            3       499.03      0.98586        0.275
>>>            3            2       698.64       3.3118       1.2812
>>>            8            5       798.45       15.237        2.863
>>>            5            3       898.26       13.897        3.557
>>>            9            5       998.06       19.535        3.557
>>>            2            1       1197.7       2.3259       2.3259
//
>! zeta12.scl
>!
>12 equal zeta tuning
> 12
>!
>99.807
>199.614
>299.422
>399.229
>499.036
>598.843
>698.650
>798.457
>898.265
>998.072
>1097.879
>1197.686
>
>...Notice the similarity...

Also...

Yahoo groups: /tuning-math/message/894 *

>15
>Gram tuning = 15.052, 4.14 cents flat
>Z tuning = 15.053, 4.26 cents flat

Ok, so following Paul's method I'll take the 5-limit
val < 15 24 35 ] and divide pairwise by log2(2 3 5),
then find the average 15.07115704285749, divide the
original val by it giving...

< 0.9952785945594528 1.5924457512951244 2.322316720638723 ]

...and then * 1200...

< 1194.3343134713434 1910.9349015541493 2786.7800647664676 ]

(Is ket notation appropriate here?  What is this, h1194.3343134713434
or h1200 or...?)

This is not apparently the Gram or the Z tuning.  To the 7-limit...

< 1195.8934635210232 1913.4295416336372
 2790.4180815490545 3348.5016978588646 ]

...this is close to the Gram tuning if I understand Gene's
nomenclature there.  Howabout the 17-limit...

< 1197.365908554304
  1915.7854536868863
  2793.8537866267093
  3352.6245439520508
  4150.868482988254
  4470.166058602735
  4869.288028120835 ]

...whoops, we blew it.

>19
>Gram tuning = 18.954, 2.93 cents sharp
>Z tuning = 18.948, 3.29 cents sharp

5-limit...

< 1202.2814046729093 1898.3390600098567 2784.2306213477896 ]

7-limit...

< 1203.8338650199978
  1900.7903131894698
  2787.8257926778892
  3358.06288663473 ]

11-limit...

< 1201.3512212496696
  1896.8703493415835
  2782.076512367656
  3351.137617170131
  4173.114768551483 ]

31-limit...

< 1201.2099597644576 //

...no cigar.

>22
>Gram tuning = 22.025, 1.35 cents flat
>Z tuning = 22.025, 1.37 cents flat

5-limit...

< 1198.7183021467067 1907.051844324306 2778.846973158275 ]

7-limit...

< 1198.6555970781733
  1906.9520862607305
  2778.7016114084927
  3378.0294099475796 ]

11-limit...

< 1198.6555970781733 //

...looks like we might have a winner here.


By the way, I've decided I like the square bracket for
ket notation, because it avoids confusion with abs. |n|,
Euclidean distance or norm or whatever ||n|| is.

-Carl


top of page bottom of page up down


Message: 9078

Date: Fri, 09 Jan 2004 14:07:17

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Paul Erlich

For an ET, just stretch so that the weighted errors of the most 
upward-biased prime and most downward-biased prime are equal in 
magnitude and opposite in sign. For 12-equal I take the mapping
[12 19 28]
divide (elementwise) by
[1 log2(3) log2(5)]
and get
[12.00000000000000  11.98766531785769  12.05894362605501]
Now we want to make the largest and smallest of these equidistant 
from 12, so we divide [12 19 28] by their average
[12.05894362605501+11.98766531785769 ]/2
giving
0.99806172487683   1.58026439772164   2.32881069137926
So Graham had the all the digits right, I just needed more precision.
Multiply by 12, and we get
1197.67406985219          1896.31727726597          2794.57282965511
Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.


           10            9       199.61       17.209       2.6508
            9            8       199.61       4.2977      0.69655
            6            5       299.42       16.223       3.3061
            5            4       399.22       12.911       2.9873
            4            3       499.03      0.98586        0.275
            3            2       698.64       3.3118       1.2812
            8            5       798.45       15.237        2.863
            5            3       898.26       13.897        3.557
            9            5       998.06       19.535        3.557
            2            1       1197.7       2.3259       2.3259
            9            4       1397.3       6.6236       1.2812
           12            5       1497.1       18.549       3.1402
            5            2       1596.9       10.585       3.1864
            8            3       1696.7       1.3401      0.29227
            3            1       1896.3       5.6377        3.557
           16            5       1996.1       17.563       2.7781
           10            3       2095.9       11.571       2.3581
           18            5       2195.7        21.86       3.3674
           15            4       2295.5       7.2733       1.2313
            4            1       2395.3       4.6519       2.3259
            9            2         2595       8.9495       2.1462
            5            1       2794.6       8.2591        3.557
           16            3       2894.4        3.666       0.6564
            6            1         3094       7.9637       3.0808
           25            4       3193.8        21.17       3.1864
           20            3       3293.6        9.245       1.5651
           15            2       3493.2       4.9473       1.0082
            8            1         3593       6.9778       2.3259
           25            3       3692.8       22.156        3.557
            9            1       3792.6       11.275        3.557
           10            1       3992.2       5.9332       1.7861
           32            3       4092.1       5.9919      0.90994
           12            1       4291.7        10.29       2.8702
           25            2       4391.5       18.844       3.3389
           27            2       4491.3       14.587       2.5348
           15            1       4690.9       2.6214      0.67097
           16            1       4790.7       9.3037       2.3259
           18            1       4990.3       13.601       3.2618
           20            1       5189.9       3.6073      0.83464
           45            2       5389.5       0.6904      0.10635
           24            1       5489.3       12.616       2.7515
           25            1       5589.1       16.518        3.557
           27            1         5689       16.913        3.557
           30            1       5888.6      0.29546     0.060214
           32            1       5988.4        11.63       2.3259
           36            1         6188       15.927       3.0808
           40            1       6387.6       1.2813      0.24076
           45            1       6587.2       3.0163      0.54924
           48            1         6687       14.941       2.6753
           50            1       6786.8       14.192       2.5146
           54            1       6886.6       19.239       3.3431
           60            1       7086.2       2.0305      0.34375
           64            1         7186       13.956       2.3259
           72            1       7385.7       18.253       2.9584
           75            1       7485.5       10.881       1.7468
           80            1       7585.3       1.0446      0.16524
           81            1       7585.3       22.551        3.557
           90            1       7784.9       5.3423      0.82292
           96            1       7884.7       17.267       2.6222
          100            1       7984.5       11.866       1.7861





--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> Gene, what do you get for the top system with the commas of 12-
equal 
> (in other words, some stretching or squashing of 12-equal)? Graham 
> seems to gave gotten pretty close below, but no cigar . . .
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> 
> wrote:
> > > Paul Erlich wrote:
> > > 
> > > > Wow. How did you find that?
> > > 
> > > Briefly (use the Reply thing so that indentation works),
> > 
> > > 22876792454961:19073486328125
> > 
> > So it was a finite search? How do you know you won't keep finding 
> > worse and worse examples if you go farther out? You might be 
> > approaching a limit, but how do you know you'll ever reach it?
> > 
> > > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > > > 
> > > > 
> > > > Shall I proceed to calculate Tenney-weighted errors for all 
> > (well, a 
> > > > bunch of) intervals? I hope you're onto something!
> > > 
> > > If you like.
> > 
> > OK, later -- gotta go perform now.
> 
> I'm back . . . Looks like you might be off in the last digit or two 
> (so maybe there is no worst comma?), but a lot of the Tenney-
weighted 
> errors are in the 3.5549 - 3.5591 range, so you're probably pretty 
> close . . . 
> 
>            10            9       199.61       17.208       2.6508
>             9            8       199.61        4.298      0.69661
>             6            5       299.42       16.223       3.3062
>             5            4       399.22        12.91       2.9872
>             4            3       499.03        0.985      0.27476
>             3            2       698.64        3.313       1.2816
>             8            5       798.45       15.238       2.8633
>             5            3       898.25       13.895       3.5566
>             9            5       998.06       19.536       3.5573
>             2            1       1197.7        2.328        2.328
>             9            4       1397.3        6.626       1.2816
>            12            5       1497.1       18.551       3.1406
>             5            2       1596.9       10.582       3.1856
>             8            3       1696.7        1.343      0.29291
>             3            1       1896.3        5.641       3.5591
>            16            5       1996.1       17.566       2.7786
>            10            3       2095.9       11.567       2.3574
>            18            5       2195.7       21.864        3.368
>            15            4       2295.5       7.2693       1.2306
>             4            1       2395.3        4.656        2.328
>             9            2         2595        8.954       2.1473
>             5            1       2794.6       8.2543       3.5549
>            16            3       2894.4        3.671       0.6573
>             6            1         3094        7.969       3.0828
>            25            4       3193.8       21.165       3.1856
>            20            3       3293.6       9.2393       1.5642
>            15            2       3493.2       4.9413        1.007
>             8            1         3593        6.984        2.328
>            25            3       3692.8        22.15        3.556
>             9            1       3792.6       11.282       3.5591
>            10            1       3992.2       5.9263        1.784
>            32            3         4092        5.999      0.91101
>            12            1       4291.7       10.297       2.8723
>            25            2       4391.5       18.837       3.3375
>            27            2       4491.3       14.595       2.5361
>            15            1       4690.9       2.6133      0.66889
>            16            1       4790.7        9.312        2.328
>            18            1       4990.3        13.61       3.2638
>            20            1       5189.9       3.5983      0.83257
>            45            2       5389.5      0.69972      0.10778
>            24            1       5489.3       12.625       2.7536
>            25            1       5589.1       16.509       3.5549
>            27            1       5688.9       16.923       3.5591
>            30            1       5888.6      0.28529      0.05814
>            32            1       5988.4        11.64        2.328
>            36            1         6188       15.938       3.0828
>            40            1       6387.6       1.2703      0.23869
>            45            1       6587.2       3.0277      0.55131
>            48            1         6687       14.953       2.6774
>            50            1       6786.8       14.181       2.5126
>            54            1       6886.6       19.251       3.3452
>            60            1       7086.2       2.0427      0.34582
>            64            1         7186       13.968        2.328
>            72            1       7385.6       18.266       2.9605
>            75            1       7485.4       10.868       1.7447
>            80            1       7585.3       1.0577      0.16731
>            81            1       7585.3       22.564       3.5591
>            90            1       7784.9       5.3557      0.82499
>            96            1       7884.7       17.281       2.6243
>           100            1       7984.5       11.853        1.784


top of page bottom of page up down


Message: 9079

Date: Fri, 09 Jan 2004 14:25:49

Subject: Re: non-1200: Tenney/heursitic meantone temperament

From: Paul Erlich

So the stretch factor is 24/(19/log2(3) + 28/log2(5)). This looks 
related to the 'dual' of the comma Graham found, but I didn't have to 
go looking for it . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> For an ET, just stretch so that the weighted errors of the most 
> upward-biased prime and most downward-biased prime are equal in 
> magnitude and opposite in sign. For 12-equal I take the mapping
> [12 19 28]
> divide (elementwise) by
> [1 log2(3) log2(5)]
> and get
> [12.00000000000000  11.98766531785769  12.05894362605501]
> Now we want to make the largest and smallest of these equidistant 
> from 12, so we divide [12 19 28] by their average
> [12.05894362605501+11.98766531785769 ]/2
> giving
> 0.99806172487683   1.58026439772164   2.32881069137926
> So Graham had the all the digits right, I just needed more 
precision.
> Multiply by 12, and we get
> 1197.67406985219          1896.31727726597          2794.57282965511
> Here it's clear we're hitting the maximum, 3.557, with both 3 and 5.
> 
> 
>            10            9       199.61       17.209       2.6508
>             9            8       199.61       4.2977      0.69655
>             6            5       299.42       16.223       3.3061
>             5            4       399.22       12.911       2.9873
>             4            3       499.03      0.98586        0.275
>             3            2       698.64       3.3118       1.2812
>             8            5       798.45       15.237        2.863
>             5            3       898.26       13.897        3.557
>             9            5       998.06       19.535        3.557
>             2            1       1197.7       2.3259       2.3259
>             9            4       1397.3       6.6236       1.2812
>            12            5       1497.1       18.549       3.1402
>             5            2       1596.9       10.585       3.1864
>             8            3       1696.7       1.3401      0.29227
>             3            1       1896.3       5.6377        3.557
>            16            5       1996.1       17.563       2.7781
>            10            3       2095.9       11.571       2.3581
>            18            5       2195.7        21.86       3.3674
>            15            4       2295.5       7.2733       1.2313
>             4            1       2395.3       4.6519       2.3259
>             9            2         2595       8.9495       2.1462
>             5            1       2794.6       8.2591        3.557
>            16            3       2894.4        3.666       0.6564
>             6            1         3094       7.9637       3.0808
>            25            4       3193.8        21.17       3.1864
>            20            3       3293.6        9.245       1.5651
>            15            2       3493.2       4.9473       1.0082
>             8            1         3593       6.9778       2.3259
>            25            3       3692.8       22.156        3.557
>             9            1       3792.6       11.275        3.557
>            10            1       3992.2       5.9332       1.7861
>            32            3       4092.1       5.9919      0.90994
>            12            1       4291.7        10.29       2.8702
>            25            2       4391.5       18.844       3.3389
>            27            2       4491.3       14.587       2.5348
>            15            1       4690.9       2.6214      0.67097
>            16            1       4790.7       9.3037       2.3259
>            18            1       4990.3       13.601       3.2618
>            20            1       5189.9       3.6073      0.83464
>            45            2       5389.5       0.6904      0.10635
>            24            1       5489.3       12.616       2.7515
>            25            1       5589.1       16.518        3.557
>            27            1         5689       16.913        3.557
>            30            1       5888.6      0.29546     0.060214
>            32            1       5988.4        11.63       2.3259
>            36            1         6188       15.927       3.0808
>            40            1       6387.6       1.2813      0.24076
>            45            1       6587.2       3.0163      0.54924
>            48            1         6687       14.941       2.6753
>            50            1       6786.8       14.192       2.5146
>            54            1       6886.6       19.239       3.3431
>            60            1       7086.2       2.0305      0.34375
>            64            1         7186       13.956       2.3259
>            72            1       7385.7       18.253       2.9584
>            75            1       7485.5       10.881       1.7468
>            80            1       7585.3       1.0446      0.16524
>            81            1       7585.3       22.551        3.557
>            90            1       7784.9       5.3423      0.82292
>            96            1       7884.7       17.267       2.6222
>           100            1       7984.5       11.866       1.7861
> 
> 
> 
> 
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > Gene, what do you get for the top system with the commas of 12-
> equal 
> > (in other words, some stretching or squashing of 12-equal)? 
Graham 
> > seems to gave gotten pretty close below, but no cigar . . .
> > 
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> 
> > wrote:
> > > > Paul Erlich wrote:
> > > > 
> > > > > Wow. How did you find that?
> > > > 
> > > > Briefly (use the Reply thing so that indentation works),
> > > 
> > > > 22876792454961:19073486328125
> > > 
> > > So it was a finite search? How do you know you won't keep 
finding 
> > > worse and worse examples if you go farther out? You might be 
> > > approaching a limit, but how do you know you'll ever reach it?
> > > 
> > > > >>TOPping it gives a narrow octave of 0.99806 2:1 octaves.
> > > > > 
> > > > > 
> > > > > Shall I proceed to calculate Tenney-weighted errors for all 
> > > (well, a 
> > > > > bunch of) intervals? I hope you're onto something!
> > > > 
> > > > If you like.
> > > 
> > > OK, later -- gotta go perform now.
> > 
> > I'm back . . . Looks like you might be off in the last digit or 
two 
> > (so maybe there is no worst comma?), but a lot of the Tenney-
> weighted 
> > errors are in the 3.5549 - 3.5591 range, so you're probably 
pretty 
> > close . . . 
> > 
> >            10            9       199.61       17.208       2.6508
> >             9            8       199.61        4.298      0.69661
> >             6            5       299.42       16.223       3.3062
> >             5            4       399.22        12.91       2.9872
> >             4            3       499.03        0.985      0.27476
> >             3            2       698.64        3.313       1.2816
> >             8            5       798.45       15.238       2.8633
> >             5            3       898.25       13.895       3.5566
> >             9            5       998.06       19.536       3.5573
> >             2            1       1197.7        2.328        2.328
> >             9            4       1397.3        6.626       1.2816
> >            12            5       1497.1       18.551       3.1406
> >             5            2       1596.9       10.582       3.1856
> >             8            3       1696.7        1.343      0.29291
> >             3            1       1896.3        5.641       3.5591
> >            16            5       1996.1       17.566       2.7786
> >            10            3       2095.9       11.567       2.3574
> >            18            5       2195.7       21.864        3.368
> >            15            4       2295.5       7.2693       1.2306
> >             4            1       2395.3        4.656        2.328
> >             9            2         2595        8.954       2.1473
> >             5            1       2794.6       8.2543       3.5549
> >            16            3       2894.4        3.671       0.6573
> >             6            1         3094        7.969       3.0828
> >            25            4       3193.8       21.165       3.1856
> >            20            3       3293.6       9.2393       1.5642
> >            15            2       3493.2       4.9413        1.007
> >             8            1         3593        6.984        2.328
> >            25            3       3692.8        22.15        3.556
> >             9            1       3792.6       11.282       3.5591
> >            10            1       3992.2       5.9263        1.784
> >            32            3         4092        5.999      0.91101
> >            12            1       4291.7       10.297       2.8723
> >            25            2       4391.5       18.837       3.3375
> >            27            2       4491.3       14.595       2.5361
> >            15            1       4690.9       2.6133      0.66889
> >            16            1       4790.7        9.312        2.328
> >            18            1       4990.3        13.61       3.2638
> >            20            1       5189.9       3.5983      0.83257
> >            45            2       5389.5      0.69972      0.10778
> >            24            1       5489.3       12.625       2.7536
> >            25            1       5589.1       16.509       3.5549
> >            27            1       5688.9       16.923       3.5591
> >            30            1       5888.6      0.28529      0.05814
> >            32            1       5988.4        11.64        2.328
> >            36            1         6188       15.938       3.0828
> >            40            1       6387.6       1.2703      0.23869
> >            45            1       6587.2       3.0277      0.55131
> >            48            1         6687       14.953       2.6774
> >            50            1       6786.8       14.181       2.5126
> >            54            1       6886.6       19.251       3.3452
> >            60            1       7086.2       2.0427      0.34582
> >            64            1         7186       13.968        2.328
> >            72            1       7385.6       18.266       2.9605
> >            75            1       7485.4       10.868       1.7447
> >            80            1       7585.3       1.0577      0.16731
> >            81            1       7585.3       22.564       3.5591
> >            90            1       7784.9       5.3557      0.82499
> >            96            1       7884.7       17.281       2.6243
> >           100            1       7984.5       11.853        1.784


top of page bottom of page up down


Message: 9081

Date: Fri, 09 Jan 2004 14:30:54

Subject: Re: Temperament agreement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > It looks like I'd be just as happy with straight lines on this 
> chart.
> 
> Could you enlighten the rest of us and give a comma list of commas 
> you want?

I wouldn't want to include any outside the 5-limit linear temperaments
having the following 18 vanishing commas. And I wouldn't mind leaving
off the last four.

81/80
32805/32768
2048/2025
15625/15552
128/125
3125/3072
250/243
78732/78125
20000/19683
25/24
648/625
135/128
256/243
393216/390625

1600000/1594323
16875/16384
2109375/2097152
531441/524288

I'm afraid I disagree with Herman about including the temperament
where the apotome (2187/2048) vanishes.

I admit I haven't heard it. My rejection is based purely on the fact
that it has errors of a similar size to others that I find marginal
(as approximations of 5-limit JI) - pelogic (135/128) and
quintuple-thirds (Blackwood's decatonic) (256/243) - while also having
about 1.5 times their complexity.

The only argument I've heard in favour of it is that Blackwood wrote
something in 21-ET that sounds good. But does it sound good because it
approximates 5-limit harmony, or despite not approximating it?


top of page bottom of page up down


Message: 9085

Date: Sat, 10 Jan 2004 00:09:40

Subject: summary -- are these right?

From: Carl Lumma

TM reduction or LLL reduction -> canonical basis

...Which of TM, LLL is preferred these days, and is there
a definition of "basis" somewhere?  It's a list of commas,
right?

----

Hermite normal form -> canonical map

...can someone give an algorithm for turning a basis (or
whatever one needs) into a map in Hermite normal form by
hand?

----

Standard val -> canonical val

...the standard val is just the best approximation of each
identity in the ET, right?  Are there any other contenders
for canonical val?

----

TOP -> weighted minimax optimum tuning -> canonical temperament

...did Gene or Graham say there's a version of TOP equivalent
to weighted rms?  And Paul, have you looked at the non-weighted
Tenney lattice?

----

Thanks,

-Carl 



________________________________________________________________________
________________________________________________________________________




------------------------------------------------------------------------
Yahoo! Groups Links

To visit your group on the web, go to:
 Yahoo groups: /tuning-math/ *

To unsubscribe from this group, send an email to:
 tuning-math-unsubscribe@xxxxxxxxxxx.xxx

Your use of Yahoo! Groups is subject to:
 Yahoo! Terms of Service *


top of page bottom of page up down


Message: 9088

Date: Sat, 10 Jan 2004 02:26:09

Subject: Re: Temperament agreement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > What I don't like about both of these proposals is the "corners" in
> > the cutoff line. I prefer straight or smoothly curved cutoffs.
> 
> It gives you the commas on your list, but you reject it anyway 

Read again.
Yahoo groups: /tuning-math/message/8521 *
I said I could live with it.

> because it doesn't make use of your personal fetish about smooth 
> curves? You may be happy to known that the constant epimercity lines 
> *are* curved on Paul's graph.
> 
> As for the rest, your obsession with curves is preposterous.

It is neither fetish, obsession nor preposterous. (I guess I asked for
that :-) And note that I said a single straight line would be fine.

But rather, it comes from an understanding of how neural nets work (as
in human perception).


top of page bottom of page up down


Message: 9091

Date: Sat, 10 Jan 2004 00:44:36

Subject: Re: also...

From: Carl Lumma

[I wrote...]

>Yahoo groups: /tuning-math/message/894 *
>
>>15
>>Gram tuning = 15.052, 4.14 cents flat
>>Z tuning = 15.053, 4.26 cents flat
>
>Ok, so following Paul's method I'll take the 5-limit
>val < 15 24 35 ] and //
>
>< 1194.3343134713434 1910.9349015541493 2786.7800647664676 ]
>
>(Is ket notation appropriate here?  What is this, h1194.3343134713434
>or h1200 or...?)
>
>This is not apparently the Gram or the Z tuning.  To the 7-limit...
>
>< 1195.8934635210232 1913.4295416336372
> 2790.4180815490545 3348.5016978588646 ]
>
>...this is close //
>
> Howabout the 17-limit...
>
>< 1197.365908554304
>  1915.7854536868863
>  2793.8537866267093
>  3352.6245439520508
>  4150.868482988254
>  4470.166058602735
>  4869.288028120835 ]
>
>...whoops, we blew it.

Maybe I shouldn't be using the standard val at limits in
which the ET is not consistent?

>>19
>>Gram tuning = 18.954, 2.93 cents sharp
>>Z tuning = 18.948, 3.29 cents sharp
>
>5-limit...
>
>< 1202.2814046729093 1898.3390600098567 2784.2306213477896 ]
>
>7-limit...
>
>< 1203.8338650199978
>  1900.7903131894698
>  2787.8257926778892
>  3358.06288663473 ]
>
>11-limit...
//
>...no cigar.

!

>>22
>>Gram tuning = 22.025, 1.35 cents flat
>>Z tuning = 22.025, 1.37 cents flat
>
>5-limit...
>
>< 1198.7183021467067 1907.051844324306 2778.846973158275 ]
>
>7-limit...
>
>< 1198.6555970781733
>  1906.9520862607305
>  2778.7016114084927
>  3378.0294099475796 ]
>
>11-limit...
>
>< 1198.6555970781733 //
>
>...looks like we might have a winner here.

Going to the 13-limit...

< 1200.7057937136167
  1910.2137627262084
  2783.454339972475
  3383.8072368292833
  4147.892741919766
  4420.780422309225 ]

...the value seems to change sharply here too.

-Carl



________________________________________________________________________
________________________________________________________________________




------------------------------------------------------------------------
Yahoo! Groups Links

To visit your group on the web, go to:
 Yahoo groups: /tuning-math/ *

To unsubscribe from this group, send an email to:
 tuning-math-unsubscribe@xxxxxxxxxxx.xxx

Your use of Yahoo! Groups is subject to:
 Yahoo! Terms of Service *


top of page bottom of page up down


Message: 9092

Date: Sun, 11 Jan 2004 21:57:09

Subject: Re: Temperament agreement

From: Paul Erlich

I don't like these two-curve boundaries when it's clear one simple 
curve could do. I personally could do without 78732/78125 and 
20000/19683, but not without 531441/524288.

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> 
> > I wouldn't want to include any outside the 5-limit linear 
temperaments
> > having the following 18 vanishing commas. And I wouldn't mind 
leaving
> > off the last four.
> 
> Your wishes can be accomodated by setting bounds for size and
> epimericity. For the short list, we have size < 93 cents and
> epimericity < 0.62, the only five limit comma which would be added 
to
> the list if we used these bounds would be 1600000/1594323. 
Presumably
> you have no objection to that, as it appears on your long list.
> 
> > 81/80
> > 32805/32768
> > 2048/2025
> > 15625/15552
> > 128/125
> > 3125/3072
> > 250/243
> > 78732/78125
> > 20000/19683
> > 25/24
> > 648/625
> > 135/128
> > 256/243
> > 393216/390625
> 
> The long list has size < 93 and epimericity < 0.68. If we were to 
use
> these bounds, we would add 6561/6250 and 20480/19683. The second of
> these, 20480/19683, has epimericity 0.6757, which is a sliver higher
> than the actual maximum epimericity of your long list, 0.6739, and 
so
> setting the bound at 0.675 would leave it off. What do you make of 
the
> 6561/6250 comma? If you had no objection to letting it on to an
> amended long list, you'd be in business there as well.
> 
> > 1600000/1594323
> > 16875/16384
> > 2109375/2097152
> > 531441/524288
> > 
> > I'm afraid I disagree with Herman about including the temperament
> > where the apotome (2187/2048) vanishes.
> 
> I'd like to see Herman's list too.


top of page bottom of page up down


Message: 9094

Date: Sun, 11 Jan 2004 21:58:50

Subject: Re: Temperament agreement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > What I don't like about both of these proposals is the "corners" 
in
> > the cutoff line. I prefer straight or smoothly curved cutoffs.
> 
> It gives you the commas on your list, but you reject it anyway 
> because it doesn't make use of your personal fetish about smooth 
> curves?

Uh-oh.

> You may be happy to known that the constant epimercity lines 
> *are* curved on Paul's graph.
> 
> As for the rest, your obsession with curves is preposterous.

It may be time to run for the hills again :)


top of page bottom of page up down


Message: 9095

Date: Sun, 11 Jan 2004 22:08:15

Subject: Re: The Two Diadiaschisma Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> These are based on the diaschisma and the diaschisma-schisma (check 
> Manuel's list if you don't believe me) of 67108864/66430125.

That's diaschisma *minus* schisma.

I've been seeing to on all the latest charts. Note its appearance as 
the "misty" comma here, connecting 12, (51,) 63, 75, and the 
excellent 87 and 99:

Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. *


> Scala 
> tells me the scale closest to diadiaschis1 in my scale archives is
> bp12_17 "12-tET approximation with minimal order 17 beats". For 
> closest to diadiaschis2 I find that it is, according to Scala, 
> exactly equidistant from duoden12 "Almost equal 12-tone subset of 
> Duodenarium".

The duodenarium is a huge Euler genus in the 5-limit lattice, with 
over 100 notes, I believe.


top of page bottom of page up down


Message: 9096

Date: Sun, 11 Jan 2004 22:13:00

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> TM reduction or LLL reduction -> canonical basis
> 
> ...Which of TM, LLL is preferred these days,

LLL is just use to "set up" for TM, I believe. TM seems like one good 
option, but there are probably better or equally good ways to define 
things beyond 2 dimensions.

> and is there
> a definition of "basis" somewhere?

You should hang it on your refrigerator. Once you do, you may be able 
to understand this: for the kernel of a temperament, it will be a 
list of linearly independent commas that don't lead to torsion; for a 
temperament, it will be a list of linearly independent intervals that 
generate the whole temperament.


> ----

> Standard val -> canonical val
> 
> ...the standard val is just the best approximation of each
> identity in the ET, right?  Are there any other contenders
> for canonical val?

Yes.

(I'm in a hurry, my apologies)


top of page bottom of page up down


Message: 9097

Date: Sun, 11 Jan 2004 22:38:32

Subject: Re: Temperament agreement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
> wrote:
> > It gives you the commas on your list, but you reject it anyway 
> > because it doesn't make use of your personal fetish about smooth 
> > curves?
> 
> Uh-oh.
> 
> > You may be happy to known that the constant epimercity lines 
> > *are* curved on Paul's graph.
> > 
> > As for the rest, your obsession with curves is preposterous.
> 
> It may be time to run for the hills again :)

Hee hee. Sorry to disappoint. :-)


top of page bottom of page up down


Message: 9098

Date: Mon, 12 Jan 2004 17:52:32

Subject: Re: summary -- are these right?

From: Carl Lumma

>> > is there a definition of "basis" somewhere?
//
>Vector Space Basis -- from MathWorld *

Ah, good.  That's what I thought.

>> >You should hang it on your refrigerator. Once you do, you may be
>> >able to understand this: for the kernel of a temperament, it will
>> >be a list of linearly independent commas that don't lead to
>> >torsion;

This is the only sense I've ever noticed it used around here, and
it's what I meant by "TM reduction -> canonical basis".

>> >for a temperament, it will be a list of linearly independent
>> >intervals that generate the whole temperament.

Generate the pitches in the temperament.  One also needs the map.

>> And did you see the posts where I compare zeta, gram, and TOP-et
>> tunings?
>
>Yup . . .

I've been wondering about working backwards from the technique
to TOP for codimension > 1 temperaments.  How would it apply to
a pair of vals?  Which commas is it tempering in the single-val
case?  etc.

-Carl


top of page bottom of page up down


Message: 9099

Date: Mon, 12 Jan 2004 18:12:05

Subject: Re: summary -- are these right?

From: Carl Lumma

>> >> And did you see the posts where I compare zeta, gram, and
>> >> TOP-et tunings?
>> >
>> >Yup . . .
>> 
>> I've been wondering about working backwards from the technique
>> to TOP for codimension > 1 temperaments.  How would it apply to
>> a pair of vals?
>
>A pair of vals -> dimension = 2. How would what apply?

We're looking for TOP for codimension 2, aren't we?

>> Which commas is it tempering in the single-val case?
>
>Nothing new to TOP here.

TOP is a single-comma technique last I heard.  Yet ETs require
more than a single comma in the 5-limit...

Oh, and just in case these got lost...

>...did Gene or Graham say there's a version of TOP equivalent
>to weighted rms?  And Paul, have you looked at the non-weighted
>Tenney lattice?

-Carl


top of page bottom of page up

Previous Next

9000 9050 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 9600 9650 9700 9750 9800 9850 9900 9950

9050 - 9075 -

top of page