TOP/RMS comparisons

Now I've got TOP optimization working for rank 2 temperaments, here
are some results of TOP and prime RMS optimizations. The first column
is the TOP period, then the RMS period, then the RMS period, then the
generators and then the errors multiplied by 1000.

The 5- and 7-limit temperaments should be the ones from Paul's paper.
For the higher limits, I had to think of something suitable.

If there's no single minimum point for the TOPs, the result is arbitrary

5-limit
Father 1185.9 1181.3 447.4 448.9 11.776 10.995
Bug 1185.8 1200.0 246.1 260.4 11.814 9.645
Dicot 1207.7 1206.4 353.2 350.5 6.383 5.912
Meantone 1201.7 1201.4 504.1 504.3 1.423 1.319
Augmented 399.0 399.0 88.5 93.1 2.451 2.000
Mavila 1206.6 1208.4 521.5 523.8 5.460 5.054
Porcupine 1196.9 1199.6 162.3 163.9 2.579 2.232
Blackwood 238.9 238.9 66.9 80.0 4.723 3.855
Dimipent 299.2 299.7 101.7 99.4 2.800 2.586
Srutal 599.6 599.4 104.7 104.8 0.742 0.696
Magic 1201.3 1201.2 380.8 380.5 1.067 0.925
Ripple 1203.3 1200.3 102.0 100.9 2.771 2.349
Hanson 1200.3 1200.2 317.1 317.1 0.245 0.228
Negripent 1201.8 1202.3 126.1 126.0 1.520 1.409
Tetracot 1201.7 1201.4 348.8 348.5 1.420 1.319
Superpyth 1197.6 1197.7 489.4 489.0 2.004 1.760
Helmholtz 1200.1 1200.1 498.3 498.3 0.060 0.048
Sensipent 1199.6 1199.9 443.0 443.0 0.345 0.297
Passion 1198.3 1197.8 98.4 98.5 1.405 1.305
Wuerschmidt 1199.7 1199.7 387.6 387.7 0.258 0.218
Compton 100.1 100.1 13.7 15.1 0.514 0.420
Amity 1199.9 1199.9 339.5 339.5 0.127 0.117
Orson 1200.2 1200.3 271.7 271.7 0.200 0.179
Vishnu 600.0 600.0 71.1 71.1 0.044 0.039
Luna 1200.0 1200.0 193.2 193.2 0.015 0.013

7-limit
Blacksmith 239.2 239.4 67.0 87.0 6.035 4.494
Dimisept 298.5 299.1 101.5 99.2 4.894 4.098
Dominant 1195.2 1195.4 495.9 496.5 3.976 3.929
August 400.0 399.1 107.3 103.8 4.893 3.944
Pajara 598.4 598.9 106.6 106.8 2.590 2.144
Semaphore 1203.7 1203.9 252.5 253.4 3.063 2.296
Meantone 1201.7 1201.2 504.1 504.0 1.423 1.151
Injera 600.9 600.7 93.6 94.5 2.986 2.615
Negrisept 1203.2 1203.5 124.8 126.0 2.661 2.146
Augene 399.0 398.8 88.5 90.5 2.451 1.856
Keemun 1203.2 1202.6 317.8 317.2 2.660 2.152
Catler 99.8 99.9 14.6 26.8 2.964 2.242
Hedgehog 598.4 599.6 162.3 164.2 2.590 2.317
Superpyth 1197.6 1197.1 489.4 488.5 2.004 1.597
Sensisept 1198.4 1199.7 443.2 443.3 1.343 1.103
Lemba 601.7 601.5 230.9 232.7 3.118 2.666
Porcupine 1196.9 1197.8 162.3 162.6 2.579 2.109
Flattone 1202.5 1203.6 507.1 507.8 2.118 1.764
Magic 1201.3 1201.1 380.8 380.7 1.067 0.895
Doublewide 599.3 600.0 272.3 274.3 2.725 2.069
Nautilus 1202.7 1202.2 83.0 82.7 2.905 2.706
Beatles 1197.1 1196.6 354.7 354.9 2.415 1.917
Liese 1202.6 1201.6 569.0 568.3 2.195 1.849
Cynder 1201.7 1200.9 232.5 232.4 1.420 1.188
Orwell 1199.5 1200.0 271.5 271.5 0.789 0.623
Garibaldi 1200.8 1200.1 498.1 498.0 0.763 0.605
Myna 1198.8 1199.3 309.9 310.0 0.977 0.793
Miracle 1200.6 1200.8 116.7 116.8 0.530 0.429
Ennealimmal 133.3 133.3 49.0 49.0 0.035 0.025

11-limit
Miracle 1200.6 1200.8 116.7 116.7 0.530 0.404
Diaschismic 599.4 599.4 103.8 103.6 1.057 0.754
Orwell 1201.2 1200.6 271.4 271.6 1.138 0.959
Shrutar 599.8 599.8 52.4 52.7 1.189 0.955
Schismic 1201.4 1200.2 497.9 497.7 1.493 1.091
Microschismic 1200.8 1200.3 498.1 498.0 0.763 0.560
Magic 1200.7 1200.1 380.9 380.7 1.401 1.022
Meantone 1201.6 1200.8 504.0 503.4 1.455 1.199
Vicentino1 1201.7 1201.2 348.8 348.8 1.422 1.225
Vicentino2 1201.7 1201.8 348.8 348.7 1.420 1.164
Mystery 41.4 41.4 16.1 16.0 0.543 0.449
Hemiennialimmal 66.7 66.7 17.6 17.6 0.042 0.032

13-limit
Mystery 41.4 41.4 16.1 15.9 0.543 0.427
Diaschismic 599.4 599.4 103.8 103.6 1.057 0.688
Cassandra1 1200.4 1200.2 498.1 498.0 0.811 0.580
Cassandra2 1201.5 1200.3 497.9 497.6 1.696 1.265

17-limit
Mystery 41.4 41.4 16.1 16.2 1.016 0.710
Diaschismic 599.4 599.6 103.9 103.7 1.127 0.741

Graham

Great. For those of us just joining in, are you using any weighting
on the RMS?

What sort of error units are these? 1.4... cents in 5-limit meantone.?
Or are these TOP damage (which I understand to be Tenney-weigthed max
error) and some kind of weighted RMS?

If I followed anything so far, it's that, for RMS, "ll-limit" means
the errors of [1 3 5 7 11] without those of 11/7 and so forth. Can
you also do arbitrary []?

-Carl

At 04:29 PM 11/27/2005, you wrote:
>Now I've got TOP optimization working for rank 2 temperaments, here
>are some results of TOP and prime RMS optimizations. The first column
>is the TOP period, then the RMS period, then the RMS period, then the
>generators and then the errors multiplied by 1000.
>
>The 5- and 7-limit temperaments should be the ones from Paul's paper.
>For the higher limits, I had to think of something suitable.
>
>If there's no single minimum point for the TOPs, the result is arbitrary
>
>5-limit
>Father 1185.9 1181.3 447.4 448.9 11.776 10.995
>Bug 1185.8 1200.0 246.1 260.4 11.814 9.645
>Dicot 1207.7 1206.4 353.2 350.5 6.383 5.912
>Meantone 1201.7 1201.4 504.1 504.3 1.423 1.319
>Augmented 399.0 399.0 88.5 93.1 2.451 2.000
>Mavila 1206.6 1208.4 521.5 523.8 5.460 5.054
>Porcupine 1196.9 1199.6 162.3 163.9 2.579 2.232
>Blackwood 238.9 238.9 66.9 80.0 4.723 3.855
>Dimipent 299.2 299.7 101.7 99.4 2.800 2.586
>Srutal 599.6 599.4 104.7 104.8 0.742 0.696
>Magic 1201.3 1201.2 380.8 380.5 1.067 0.925
>Ripple 1203.3 1200.3 102.0 100.9 2.771 2.349
>Hanson 1200.3 1200.2 317.1 317.1 0.245 0.228
>Negripent 1201.8 1202.3 126.1 126.0 1.520 1.409
>Tetracot 1201.7 1201.4 348.8 348.5 1.420 1.319
>Superpyth 1197.6 1197.7 489.4 489.0 2.004 1.760
>Helmholtz 1200.1 1200.1 498.3 498.3 0.060 0.048
>Sensipent 1199.6 1199.9 443.0 443.0 0.345 0.297
>Passion 1198.3 1197.8 98.4 98.5 1.405 1.305
>Wuerschmidt 1199.7 1199.7 387.6 387.7 0.258 0.218
>Compton 100.1 100.1 13.7 15.1 0.514 0.420
>Amity 1199.9 1199.9 339.5 339.5 0.127 0.117
>Orson 1200.2 1200.3 271.7 271.7 0.200 0.179
>Vishnu 600.0 600.0 71.1 71.1 0.044 0.039
>Luna 1200.0 1200.0 193.2 193.2 0.015 0.013
>
>7-limit
>Blacksmith 239.2 239.4 67.0 87.0 6.035 4.494
>Dimisept 298.5 299.1 101.5 99.2 4.894 4.098
>Dominant 1195.2 1195.4 495.9 496.5 3.976 3.929
>August 400.0 399.1 107.3 103.8 4.893 3.944
>Pajara 598.4 598.9 106.6 106.8 2.590 2.144
>Semaphore 1203.7 1203.9 252.5 253.4 3.063 2.296
>Meantone 1201.7 1201.2 504.1 504.0 1.423 1.151
>Injera 600.9 600.7 93.6 94.5 2.986 2.615
>Negrisept 1203.2 1203.5 124.8 126.0 2.661 2.146
>Augene 399.0 398.8 88.5 90.5 2.451 1.856
>Keemun 1203.2 1202.6 317.8 317.2 2.660 2.152
>Catler 99.8 99.9 14.6 26.8 2.964 2.242
>Hedgehog 598.4 599.6 162.3 164.2 2.590 2.317
>Superpyth 1197.6 1197.1 489.4 488.5 2.004 1.597
>Sensisept 1198.4 1199.7 443.2 443.3 1.343 1.103
>Lemba 601.7 601.5 230.9 232.7 3.118 2.666
>Porcupine 1196.9 1197.8 162.3 162.6 2.579 2.109
>Flattone 1202.5 1203.6 507.1 507.8 2.118 1.764
>Magic 1201.3 1201.1 380.8 380.7 1.067 0.895
>Doublewide 599.3 600.0 272.3 274.3 2.725 2.069
>Nautilus 1202.7 1202.2 83.0 82.7 2.905 2.706
>Beatles 1197.1 1196.6 354.7 354.9 2.415 1.917
>Liese 1202.6 1201.6 569.0 568.3 2.195 1.849
>Cynder 1201.7 1200.9 232.5 232.4 1.420 1.188
>Orwell 1199.5 1200.0 271.5 271.5 0.789 0.623
>Garibaldi 1200.8 1200.1 498.1 498.0 0.763 0.605
>Myna 1198.8 1199.3 309.9 310.0 0.977 0.793
>Miracle 1200.6 1200.8 116.7 116.8 0.530 0.429
>Ennealimmal 133.3 133.3 49.0 49.0 0.035 0.025
>
>11-limit
>Miracle 1200.6 1200.8 116.7 116.7 0.530 0.404
>Diaschismic 599.4 599.4 103.8 103.6 1.057 0.754
>Orwell 1201.2 1200.6 271.4 271.6 1.138 0.959
>Shrutar 599.8 599.8 52.4 52.7 1.189 0.955
>Schismic 1201.4 1200.2 497.9 497.7 1.493 1.091
>Microschismic 1200.8 1200.3 498.1 498.0 0.763 0.560
>Magic 1200.7 1200.1 380.9 380.7 1.401 1.022
>Meantone 1201.6 1200.8 504.0 503.4 1.455 1.199
>Vicentino1 1201.7 1201.2 348.8 348.8 1.422 1.225
>Vicentino2 1201.7 1201.8 348.8 348.7 1.420 1.164
>Mystery 41.4 41.4 16.1 16.0 0.543 0.449
>Hemiennialimmal 66.7 66.7 17.6 17.6 0.042 0.032
>
>13-limit
>Mystery 41.4 41.4 16.1 15.9 0.543 0.427
>Diaschismic 599.4 599.4 103.8 103.6 1.057 0.688
>Cassandra1 1200.4 1200.2 498.1 498.0 0.811 0.580
>Cassandra2 1201.5 1200.3 497.9 497.6 1.696 1.265
>
>17-limit
>Mystery 41.4 41.4 16.1 16.2 1.016 0.710
>Diaschismic 599.4 599.6 103.9 103.7 1.127 0.741

On 11/27/05, Carl Lumma <ekin@lumma.org> wrote:
> Great. For those of us just joining in, are you using any weighting
> on the RMS?

Yes, the same as for TOP.

> What sort of error units are these? 1.4... cents in 5-limit meantone.?
> Or are these TOP damage (which I understand to be Tenney-weigthed max
> error) and some kind of weighted RMS?

They're both Tenney weighted error, which is dimensionless, multiplied by 1000.

> If I followed anything so far, it's that, for RMS, "ll-limit" means
> the errors of [1 3 5 7 11] without those of 11/7 and so forth. Can
> you also do arbitrary []?

Yes to the first. You can have arbitrary primes, or a composite like
9 instead of 3. To have arbitrary intervals you'd have to change the
code because it assumes the size of the intervals is the same as their
weight. That makes it cleaner mathematically. The result about the
optimal RMS matching the standard deviation wouldn't work any more but
I don't think the code would be much more complex.

Graham

Good on all counts.

-Carl

At 12:29 AM 11/29/2005, you wrote:
>On 11/27/05, Carl Lumma <ekin@lumma.org> wrote:
>> Great. For those of us just joining in, are you using any weighting
>> on the RMS?
>
>Yes, the same as for TOP.
>
>> What sort of error units are these? 1.4... cents in 5-limit meantone.?
>> Or are these TOP damage (which I understand to be Tenney-weigthed max
>> error) and some kind of weighted RMS?
>
>They're both Tenney weighted error, which is dimensionless, multiplied by 1000.
>
>> If I followed anything so far, it's that, for RMS, "ll-limit" means
>> the errors of [1 3 5 7 11] without those of 11/7 and so forth. Can
>> you also do arbitrary []?
>
>Yes to the first. You can have arbitrary primes, or a composite like
>9 instead of 3. To have arbitrary intervals you'd have to change the
>code because it assumes the size of the intervals is the same as their
>weight. That makes it cleaner mathematically. The result about the
>optimal RMS matching the standard deviation wouldn't work any more but
>I don't think the code would be much more complex.
>
>
> Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> On 11/27/05, Carl Lumma <ekin@l...> wrote:
> > Great. For those of us just joining in, are you using any
weighting
> > on the RMS?
>
> Yes, the same as for TOP.

So this is PORMSWE or whatever else you called it. Do we need three
different acronyms for the same thing?

> > What sort of error units are these? 1.4... cents in 5-limit
meantone.?
> > Or are these TOP damage (which I understand to be Tenney-weigthed
max
> > error) and some kind of weighted RMS?
>
> They're both Tenney weighted error, which is dimensionless,
>multiplied by 1000.
>
> > If I followed anything so far, it's that, for RMS, "ll-limit"
means
> > the errors of [1 3 5 7 11] without those of 11/7 and so forth.
Can
> > you also do arbitrary []?
>
> Yes to the first.

I don't think that's right. You have to substitute "2" for "1", so it
should be {2 3 5 7 11} -- right Graham? Otherwise, there's no way the
octaves could get tempered, but they clearly did.

I think it's worth emphasizing the similarities with TOP here; in a
certain sense, the error of 11/7 does implicitly get considered with
weight 1/log(77) in both schemes.

Carl,

Graham was too hasty in approving of your [1 3 5 7 11]. He should
have insisted you change '1' to '2', as I just tried to explain in
another post.

-Paul

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> Good on all counts.
>
> -Carl
>
> At 12:29 AM 11/29/2005, you wrote:
> >On 11/27/05, Carl Lumma <ekin@l...> wrote:
> >> Great. For those of us just joining in, are you using any
weighting
> >> on the RMS?
> >
> >Yes, the same as for TOP.
> >
> >> What sort of error units are these? 1.4... cents in 5-limit
meantone.?
> >> Or are these TOP damage (which I understand to be Tenney-
weigthed max
> >> error) and some kind of weighted RMS?
> >
> >They're both Tenney weighted error, which is dimensionless,
multiplied by 1000.
> >
> >> If I followed anything so far, it's that, for RMS, "ll-limit"
means
> >> the errors of [1 3 5 7 11] without those of 11/7 and so forth.
Can
> >> you also do arbitrary []?
> >
> >Yes to the first. You can have arbitrary primes, or a composite
like
> >9 instead of 3. To have arbitrary intervals you'd have to change
the
> >code because it assumes the size of the intervals is the same as
their
> >weight. That makes it cleaner mathematically. The result about
the
> >optimal RMS matching the standard deviation wouldn't work any more
but
> >I don't think the code would be much more complex.
> >
> >
> > Graham
>

On 11/30/05, Paul Erlich <perlich@aya.yale.edu> wrote:

> So this is PORMSWE or whatever else you called it. Do we need three
> different acronyms for the same thing?

No, but PORMSWE's a mouthful and needs to be explained anyway so I
don't use it much

> I don't think that's right. You have to substitute "2" for "1", so it
> should be {2 3 5 7 11} -- right Graham? Otherwise, there's no way the
> octaves could get tempered, but they clearly did.

Yes, although there is an octave equivalent formalism.

> I think it's worth emphasizing the similarities with TOP here; in a
> certain sense, the error of 11/7 does implicitly get considered with
> weight 1/log(77) in both schemes.

They're both averages. The rationale for the implicit result isn't as
strong as for TOP, but the idea is that it's a simplification of an
arbitrary prime limited set.

Graham

>Carl,
>
>Graham was too hasty in approving of your [1 3 5 7 11]. He should
>have insisted you change '1' to '2', as I just tried to explain in
>another post.
>
>-Paul

Yes, I saw that. An important detail, but not exactly material
to what I was asking him.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> On 11/30/05, Paul Erlich <perlich@a...> wrote:
>
> > So this is PORMSWE or whatever else you called it. Do we need
three
> > different acronyms for the same thing?
>
> No, but PORMSWE's a mouthful and needs to be explained anyway so I
> don't use it much
>
> > I don't think that's right. You have to substitute "2" for "1",
so it
> > should be {2 3 5 7 11} -- right Graham? Otherwise, there's no way
the
> > octaves could get tempered, but they clearly did.
>
> Yes, although there is an octave equivalent formalism.

How does that go?

> > I think it's worth emphasizing the similarities with TOP here; in
a
> > certain sense, the error of 11/7 does implicitly get considered
with
> > weight 1/log(77) in both schemes.
>
> They're both averages. The rationale for the implicit result isn't
as
> strong as for TOP,

Well, you made some arguments and conjectures and they may very well
be awfully close!

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >Carl,
> >
> >Graham was too hasty in approving of your [1 3 5 7 11]. He should
> >have insisted you change '1' to '2', as I just tried to explain in
> >another post.
> >
> >-Paul
>
> Yes, I saw that. An important detail, but not exactly material
> to what I was asking him.
>
> -Carl

It's important because it implies that 11/7 is going to be implicitly
weighted a lot more than 77/64.

>> >Carl,
>> >
>> >Graham was too hasty in approving of your [1 3 5 7 11]. He should
>> >have insisted you change '1' to '2', as I just tried to explain in
>> >another post.
>> >
>> >-Paul
>>
>> Yes, I saw that. An important detail, but not exactly material
>> to what I was asking him.
>>
>> -Carl
>
>It's important because it implies that 11/7 is going to be implicitly
>weighted a lot more than 77/64.

Yes, I know, but again it has nothing to do with what I was asking
him, which was the only reason I concocted the example.

-Carl

They're closer for more accurate temperaments, as I'd expect.
The only points where there's more difference than I'd
expect are the 5-limit porcupine period and the 5-limit
blackwood generator. Do you know what's causing the difference
in these cases (lack of a single TOP minimum)?

How does Paul deal with the lack of a single minimum in his
paper (I assume this never happens with RMS)?

-Carl

>At 04:29 PM 11/27/2005, you wrote:
>>Now I've got TOP optimization working for rank 2 temperaments, here
>>are some results of TOP and prime RMS optimizations. The first column
>>is the TOP period, then the RMS period, then the RMS period, then the
>>generators and then the errors multiplied by 1000.
>>
>>The 5- and 7-limit temperaments should be the ones from Paul's paper.
>>For the higher limits, I had to think of something suitable.
>>
>>If there's no single minimum point for the TOPs, the result is arbitrary
>>
>>5-limit
>>Father 1185.9 1181.3 447.4 448.9 11.776 10.995
>>Bug 1185.8 1200.0 246.1 260.4 11.814 9.645
>>Dicot 1207.7 1206.4 353.2 350.5 6.383 5.912
>>Meantone 1201.7 1201.4 504.1 504.3 1.423 1.319
>>Augmented 399.0 399.0 88.5 93.1 2.451 2.000
>>Mavila 1206.6 1208.4 521.5 523.8 5.460 5.054
>>Porcupine 1196.9 1199.6 162.3 163.9 2.579 2.232
>>Blackwood 238.9 238.9 66.9 80.0 4.723 3.855
>>Dimipent 299.2 299.7 101.7 99.4 2.800 2.586
>>Srutal 599.6 599.4 104.7 104.8 0.742 0.696
>>Magic 1201.3 1201.2 380.8 380.5 1.067 0.925
>>Ripple 1203.3 1200.3 102.0 100.9 2.771 2.349
>>Hanson 1200.3 1200.2 317.1 317.1 0.245 0.228
>>Negripent 1201.8 1202.3 126.1 126.0 1.520 1.409
>>Tetracot 1201.7 1201.4 348.8 348.5 1.420 1.319
>>Superpyth 1197.6 1197.7 489.4 489.0 2.004 1.760
>>Helmholtz 1200.1 1200.1 498.3 498.3 0.060 0.048
>>Sensipent 1199.6 1199.9 443.0 443.0 0.345 0.297
>>Passion 1198.3 1197.8 98.4 98.5 1.405 1.305
>>Wuerschmidt 1199.7 1199.7 387.6 387.7 0.258 0.218
>>Compton 100.1 100.1 13.7 15.1 0.514 0.420
>>Amity 1199.9 1199.9 339.5 339.5 0.127 0.117
>>Orson 1200.2 1200.3 271.7 271.7 0.200 0.179
>>Vishnu 600.0 600.0 71.1 71.1 0.044 0.039
>>Luna 1200.0 1200.0 193.2 193.2 0.015 0.013
>>
>>7-limit
>>Blacksmith 239.2 239.4 67.0 87.0 6.035 4.494
>>Dimisept 298.5 299.1 101.5 99.2 4.894 4.098
>>Dominant 1195.2 1195.4 495.9 496.5 3.976 3.929
>>August 400.0 399.1 107.3 103.8 4.893 3.944
>>Pajara 598.4 598.9 106.6 106.8 2.590 2.144
>>Semaphore 1203.7 1203.9 252.5 253.4 3.063 2.296
>>Meantone 1201.7 1201.2 504.1 504.0 1.423 1.151
>>Injera 600.9 600.7 93.6 94.5 2.986 2.615
>>Negrisept 1203.2 1203.5 124.8 126.0 2.661 2.146
>>Augene 399.0 398.8 88.5 90.5 2.451 1.856
>>Keemun 1203.2 1202.6 317.8 317.2 2.660 2.152
>>Catler 99.8 99.9 14.6 26.8 2.964 2.242
>>Hedgehog 598.4 599.6 162.3 164.2 2.590 2.317
>>Superpyth 1197.6 1197.1 489.4 488.5 2.004 1.597
>>Sensisept 1198.4 1199.7 443.2 443.3 1.343 1.103
>>Lemba 601.7 601.5 230.9 232.7 3.118 2.666
>>Porcupine 1196.9 1197.8 162.3 162.6 2.579 2.109
>>Flattone 1202.5 1203.6 507.1 507.8 2.118 1.764
>>Magic 1201.3 1201.1 380.8 380.7 1.067 0.895
>>Doublewide 599.3 600.0 272.3 274.3 2.725 2.069
>>Nautilus 1202.7 1202.2 83.0 82.7 2.905 2.706
>>Beatles 1197.1 1196.6 354.7 354.9 2.415 1.917
>>Liese 1202.6 1201.6 569.0 568.3 2.195 1.849
>>Cynder 1201.7 1200.9 232.5 232.4 1.420 1.188
>>Orwell 1199.5 1200.0 271.5 271.5 0.789 0.623
>>Garibaldi 1200.8 1200.1 498.1 498.0 0.763 0.605
>>Myna 1198.8 1199.3 309.9 310.0 0.977 0.793
>>Miracle 1200.6 1200.8 116.7 116.8 0.530 0.429
>>Ennealimmal 133.3 133.3 49.0 49.0 0.035 0.025

Carl Lumma wrote:
> They're closer for more accurate temperaments, as I'd expect.
> The only points where there's more difference than I'd
> expect are the 5-limit porcupine period and the 5-limit
> blackwood generator. Do you know what's causing the difference
> in these cases (lack of a single TOP minimum)?

For blackwood, it's because there's no single TOP(max) minimum. The error in 3 is independent of the generator. I don't know what's happening with Porcupine.

> How does Paul deal with the lack of a single minimum in his
> paper (I assume this never happens with RMS)?

I don't know, but his blackwood optimum is close to what I give for its RMS optimum.

Yes, the RMS always has a well defined minimum except for true rank 2 temperaments. An untrue temperament would be an equal temperament masquerading as a higher rank temperament. Then, no consonances depend on the generator, so the tuning is arbitrary.

Graham

> -Carl
> > >>At 04:29 PM 11/27/2005, you wrote:
>>
>>>Now I've got TOP optimization working for rank 2 temperaments, here
>>>are some results of TOP and prime RMS optimizations. The first column
>>>is the TOP period, then the RMS period, then the RMS period, then the
>>>generators and then the errors multiplied by 1000.
>>>
>>>The 5- and 7-limit temperaments should be the ones from Paul's paper. >>>For the higher limits, I had to think of something suitable.
>>>
>>>If there's no single minimum point for the TOPs, the result is arbitrary
>>>
>>>5-limit
>>>Father 1185.9 1181.3 447.4 448.9 11.776 10.995
>>>Bug 1185.8 1200.0 246.1 260.4 11.814 9.645
>>>Dicot 1207.7 1206.4 353.2 350.5 6.383 5.912
>>>Meantone 1201.7 1201.4 504.1 504.3 1.423 1.319
>>>Augmented 399.0 399.0 88.5 93.1 2.451 2.000
>>>Mavila 1206.6 1208.4 521.5 523.8 5.460 5.054
>>>Porcupine 1196.9 1199.6 162.3 163.9 2.579 2.232
>>>Blackwood 238.9 238.9 66.9 80.0 4.723 3.855
>>>Dimipent 299.2 299.7 101.7 99.4 2.800 2.586
>>>Srutal 599.6 599.4 104.7 104.8 0.742 0.696
>>>Magic 1201.3 1201.2 380.8 380.5 1.067 0.925
>>>Ripple 1203.3 1200.3 102.0 100.9 2.771 2.349
>>>Hanson 1200.3 1200.2 317.1 317.1 0.245 0.228
>>>Negripent 1201.8 1202.3 126.1 126.0 1.520 1.409
>>>Tetracot 1201.7 1201.4 348.8 348.5 1.420 1.319
>>>Superpyth 1197.6 1197.7 489.4 489.0 2.004 1.760
>>>Helmholtz 1200.1 1200.1 498.3 498.3 0.060 0.048
>>>Sensipent 1199.6 1199.9 443.0 443.0 0.345 0.297
>>>Passion 1198.3 1197.8 98.4 98.5 1.405 1.305
>>>Wuerschmidt 1199.7 1199.7 387.6 387.7 0.258 0.218
>>>Compton 100.1 100.1 13.7 15.1 0.514 0.420
>>>Amity 1199.9 1199.9 339.5 339.5 0.127 0.117
>>>Orson 1200.2 1200.3 271.7 271.7 0.200 0.179
>>>Vishnu 600.0 600.0 71.1 71.1 0.044 0.039
>>>Luna 1200.0 1200.0 193.2 193.2 0.015 0.013