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Message: 9075 - Contents - Hide Contents Date: Fri, 09 Jan 2004 21:16:36 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>> >o 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
Message: 9076 - Contents - Hide Contents 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? > > -CarlSeemingly -- 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 . . .
Message: 9077 - Contents - Hide Contents Date: Fri, 09 Jan 2004 23:57:30 Subject: also... From: Carl Lumma>>> >0 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 * [with cont.] >15 >Gram tuning = 15.052, 4.14 cents flat >Z tuning = 15.053, 4.26 cents flatOk, 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
Message: 9078 - Contents - Hide Contents
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
Message: 9079 - Contents - Hide Contents
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
Message: 9081 - Contents - Hide Contents
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?
Message: 9082 - Contents - Hide Contents
Date: Fri, 09 Jan 2004 19:26:53
Subject: Re: non-1200: Tenney/heursitic meantone temperament
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> right, right . . . just wanted to do a cross-check between you and
> graham . . . so what's the formula for top linear in 7-limit (for
> pajara and/or in general)?
I'm not using a formula, I'm finding points in a subpace of the val
space closest to JIP. This may give a simpilical region, in which
case we can find the centroid by taking the average of the points
(remember, we are in a vector space.) Another way to look at it is
that we could replace the Tenney norm with this:
|| |e2 e3 ... ep> ||_a =
(|e2|^a + |e3 log2(3)|^a ... + |ep log2(p)|^a)^(1/a)
If 1/a+1/b=1, the dual space for that norm would be
|| <f2 f3 ... fp> || =
(|f2|^b + |f3/log2(3)|^b + ... + |fp log2(p)}^b}^(1/b)
As a-->1, b--> infinity, and the point we want would be the limit of
this. I wouldn't recommend it as a computation method.
Message: 9083 - Contents - Hide Contents
Date: Fri, 09 Jan 2004 19:28:57
Subject: Re: non-1200: Tenney/heursitic meantone temperament
From: Gene Ward Smith
--- 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?
Or combining vals. A set of commas determines a subspace of the val
space, the null space. A set of vals determines a subspace, the span.
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Message: 9084 - Contents - Hide Contents
Date: Sat, 10 Jan 2004 19:56:40
Subject: Norms on wedge products
From: Gene Ward Smith
We have a basis for these in terms of m-fold wedge products of p-adic
valuations; so a coordinate will look like
c vp1^vp2^...^vpm
for some subset of m primes in our limit, arranged in increasing order.
We get a norm consistent with the Tenney norm by taking the maximum of
the quantities
| c / (log2(p1) * log2(p2) ... log2(pm)) |
So, for instance, if <<w1 w2 w3 w4 w5 w6|| is a 7-limit wedgie, the
norm for it would be
|| <<w1 w2 w3 w4 w5 w6|| || = Max(|w1/q3}, |w2/q5|, |w3/q7|,
|w4/(q3 * q5)|, |w5/(q3 * q7)|, |w6/(q5 * q7)|)
where q3=log2(3), etc.
Message: 9085 - Contents - Hide Contents
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
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Message: 9087 - Contents - Hide Contents
Date: Sat, 10 Jan 2004 00:48:02
Subject: Re: Temperament agreement
From: Gene Ward Smith
--- 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? 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.
Message: 9088 - Contents - Hide Contents
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 * [with cont.]
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).
Message: 9089 - Contents - Hide Contents
Date: Sat, 10 Jan 2004 02:29:59
Subject: 5-limit comma list
From: Gene Ward Smith
These are all the 5-limit commas with size less than 100 cents and
epimericity less than 2/3; it looks reasonable to me.
135/128 Pelogic
256/243 Blackwood
6561/6250 Ragitonic/Diaschizmoid
25/24 Dicot
648/625 Diminished
16875/16384 Negri
250/243 Porcupine
128/125 Augmented
3125/3072 Magic
20000/19683 Tetracot
81/80 Meantone
2048/2025 Diaschismic
78732/78125 Hemisixths
393216/390625 Wuerschmidt
2109375/2097152 Orwell/Semicomma
15625/15552 Kleismic
1600000/1594323 Amity
32805/32768 Schismic
Message: 9090 - Contents - Hide Contents
Date: Sat, 10 Jan 2004 04:08:49
Subject: Error and complexity Tenney style
From: Gene Ward Smith
We have an error measurement already; a temperament corresonds to a
subspace of the val space, and its error is its minimum distance to
the JIP.
For complexity maybe we want to put a measure on wedgies products.
For example, for a 7-limit wedgie l = [l1 l2 l3 l4 l5 l6] and a monzo
|w1 w2 w3 w4> we can take a wedge product and get a val
<l3*w4+l2*w3+w2*l1, l5*w4+l4*w3-w1*l1,
w4*l6-l4*w2-w1*l2, -w3*l6-l5*w2-w1*l3|
We can use all the same formulas for arbitary points in the Tenney
space. Suppose we wedge with all unit vectors, which is to say,
with w = |w1 w2 w3 w4> such that
||w|| = |w1|+log2(3)|w2|+log2(5)|w3|+log2(7)|w4| = 1
The maximum of the norms of all the corresponding points in the
Tenney space when wedged with all w of norm 1 will give us a norm on
l. We can also start from the val side, and get another norm on wedge
products.
Message: 9091 - Contents - Hide Contents
Date: Sat, 10 Jan 2004 00:44:36
Subject: Re: also...
From: Carl Lumma
[I wrote...]
>> 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
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Message: 9092 - Contents - Hide Contents
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.
Message: 9093 - Contents - Hide Contents
Date: Sun, 11 Jan 2004 06:44:27
Subject: The Two Diadiaschisma Scales
From: Gene Ward Smith
These are based on the diaschisma and the diaschisma-schisma (check
Manuel's list if you don't believe me) of 67108864/66430125. 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". These scales seem to be warping into some sort of
circulating temperament.
! diadiaschis1.scl
Diadiaschisma scale 2048/2025 67108864/66430125
12
!
135/128
18225/16384
1215/1024
512/405
4/3
45/32
3/2
405/256
54675/32768
16/9
256/135
2
! diadiaschis2.scl
Diadiaschisma scale 2048/2025 67108864/66430125
12
!
135/128
9/8
1215/1024
512/405
4/3
64/45
3/2
405/256
54675/32768
32768/18225
256/135
2
Message: 9094 - Contents - Hide Contents
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 :)
Message: 9095 - Contents - Hide Contents
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-temp... * [with cont.] (Wayb.)
> 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.
Message: 9096 - Contents - Hide Contents
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)
Message: 9097 - Contents - Hide Contents
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. :-)
Message: 9098 - Contents - Hide Contents
Date: Mon, 12 Jan 2004 17:52:32
Subject: Re: summary -- are these right?
From: Carl Lumma
>>> >s there a definition of "basis" somewhere?
//
>Vector Space Basis -- from MathWorld * [with cont.]
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
Message: 9099 - Contents - Hide Contents
Date: Mon, 12 Jan 2004 18:12:05
Subject: Re: summary -- are these right?
From: Carl Lumma
>>>> >nd 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...