Even more ridiculous 5-comma list

I merged the list I gave previously with everything which came up as a TM basis element in the previous posting, and looked at the resulting list. The numbers are unweighted complexity and badness, followed by weighted complexity and badness. My attempt to name
2^(-90) 3^(-15) 5^49 was not well received before, but it remains a remarkable comma; so also is 2^161 3^(-84) 5^(-12).

5/3 .816497 340.389827 .871098 413.347156

4/3 .816497 191.697610 .694533 117.986270

6/5 .816497 121.490388 .871098 147.529986

9/8 1.632993 313.939984 1.389065 193.224150

10/9 1.414214 210.621665 1.312845 168.499608

27/25 2.160247 358.982164 2.117162 337.928683

16/15 1.414214 129.016176 1.312845 103.214335

135/128 2.943920 461.234866 2.558773 302.858095

25/24 1.414214 81.605488 1.597771 117.684239

648/625 3.265986 385.301392 3.484393 467.884825

250/243 3.559026 359.557053 3.413659 317.274064

128/125 2.449490 142.232058 2.613295 172.717315

3125/3072 3.741657 239.363598 4.128051 321.440927

81/80 2.943920 107.611082 2.558773 70.660069

2048/2025 4.320494 210.722021 3.822599 145.943888

78732/78125 6.683313 345.537861 6.772338 359.530849

393216/390625 6.164414 251.101816 6.722154 325.611578

2109375/2097152 7.257180 305.925879 7.187007 297.136920

15625/15552 4.546061 96.735253 4.990527 127.973025

1600000/1594323 9.273618 305.537220 8.314888 220.234641

(2)^8*(3)^14/(5)^13 11.045361 372.731488 11.618628 433.831341

(2)^7*(3)^41/(5)^31 30.232433 2337.628278 30.446552 2387.649135

(2)^23*(3)^6/(5)^14 9.933110 190.150656 11.205944 273.015594

(5)^19/(2)^14/(3)^19 15.513435 391.217013 16.550868 475.068368

(3)^8*(5)/(2)^15 6.976150 54.895845 5.957336 34.186002

(5)^34/(2)^52/(3)^17 24.041631 477.611283 27.162114 688.768879

(2)^38/(3)^2/(5)^15 13.140269 137.999227 13.679676 155.700958

(3)^35/(2)^16/(5)^17 24.752104 386.194094 22.873419 304.763201

(2)*(5)^18/(3)^27 19.442222 188.083834 19.054459 177.053149

(2)^91/(3)^12/(5)^31 31.379399 463.252410 31.274979 458.643165

(3)^10*(5)^16/(2)^53 18.547237 113.091251 17.879417 101.309796

(2)^37*(3)^25/(5)^33 24.344746 178.739193 27.016133 244.271942

(5)^51/(2)^36/(3)^52 42.055519 346.619485 44.707429 416.411830

(2)^54*(5)^2/(3)^37 29.427878 146.241253 25.054172 90.247095

(3)^47*(5)^14/(2)^107 45.188494 326.880658 39.136911 212.356350

(2)^144/(3)^22/(5)^47 49.846431 352.012120 49.076087 335.942685

(3)^62/(2)^17/(5)^35 43.962105 256.793793 41.677831 218.808617

(3)^146/(2)^178/(5)^23 111.016515 1530.538820 95.204576 965.283848

(2)^92*(3)^191/(5)^170 148.123822 943.596569 154.481332 1070.384081

(2)^71*(5)^37/(3)^99 70.743669 181.089163 63.242667 129.377945

(5)^49/(2)^90/(3)^15 35.505868 34.075240 39.697565 47.624467

(5)^256/(2)^111/(3)^305 231.633331 1286.017248 238.618149 1405.898784

(3)^153*(5)^73/(2)^412 163.109370 226.729751 143.955865 155.869043

(2)^161/(3)^84/(5)^12 73.972968 48.546279 63.238794 30.331058

(2)^21*(3)^290/(5)^207 211.217108 235.136329 210.059812 231.292406

(2)^140*(5)^195/(3)^374 264.538592 419.006551 247.279087 342.228257

(3)^237*(5)^85/(2)^573 235.992938 74.673031 205.656081 49.418713

Hi Gene,

Did you ever look at my list here:
http://www.kees.cc/tuning/s235.html

Kees

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@juno.com>
To: <tuning-math@yahoogroups.com>
Sent: Monday, November 25, 2002 6:48 AM
Subject: [tuning-math] Even more ridiculous 5-comma list

I merged the list I gave previously with everything which came up as a TM
basis element in the previous posting, and looked at the resulting list. The
numbers are unweighted complexity and badness, followed by weighted
complexity and badness. My attempt to name
2^(-90) 3^(-15) 5^49 was not well received before, but it remains a
remarkable comma; so also is 2^161 3^(-84) 5^(-12).

5/3 .816497 340.389827 .871098 413.347156

4/3 .816497 191.697610 .694533 117.986270

6/5 .816497 121.490388 .871098 147.529986

9/8 1.632993 313.939984 1.389065 193.224150

10/9 1.414214 210.621665 1.312845 168.499608

27/25 2.160247 358.982164 2.117162 337.928683

16/15 1.414214 129.016176 1.312845 103.214335

135/128 2.943920 461.234866 2.558773 302.858095

25/24 1.414214 81.605488 1.597771 117.684239

648/625 3.265986 385.301392 3.484393 467.884825

250/243 3.559026 359.557053 3.413659 317.274064

128/125 2.449490 142.232058 2.613295 172.717315

3125/3072 3.741657 239.363598 4.128051 321.440927

81/80 2.943920 107.611082 2.558773 70.660069

2048/2025 4.320494 210.722021 3.822599 145.943888

78732/78125 6.683313 345.537861 6.772338 359.530849

393216/390625 6.164414 251.101816 6.722154 325.611578

2109375/2097152 7.257180 305.925879 7.187007 297.136920

15625/15552 4.546061 96.735253 4.990527 127.973025

1600000/1594323 9.273618 305.537220 8.314888 220.234641

(2)^8*(3)^14/(5)^13 11.045361 372.731488 11.618628 433.831341

(2)^7*(3)^41/(5)^31 30.232433 2337.628278 30.446552 2387.649135

(2)^23*(3)^6/(5)^14 9.933110 190.150656 11.205944 273.015594

(5)^19/(2)^14/(3)^19 15.513435 391.217013 16.550868 475.068368

(3)^8*(5)/(2)^15 6.976150 54.895845 5.957336 34.186002

(5)^34/(2)^52/(3)^17 24.041631 477.611283 27.162114 688.768879

(2)^38/(3)^2/(5)^15 13.140269 137.999227 13.679676 155.700958

(3)^35/(2)^16/(5)^17 24.752104 386.194094 22.873419 304.763201

(2)*(5)^18/(3)^27 19.442222 188.083834 19.054459 177.053149

(2)^91/(3)^12/(5)^31 31.379399 463.252410 31.274979 458.643165

(3)^10*(5)^16/(2)^53 18.547237 113.091251 17.879417 101.309796

(2)^37*(3)^25/(5)^33 24.344746 178.739193 27.016133 244.271942

(5)^51/(2)^36/(3)^52 42.055519 346.619485 44.707429 416.411830

(2)^54*(5)^2/(3)^37 29.427878 146.241253 25.054172 90.247095

(3)^47*(5)^14/(2)^107 45.188494 326.880658 39.136911 212.356350

(2)^144/(3)^22/(5)^47 49.846431 352.012120 49.076087 335.942685

(3)^62/(2)^17/(5)^35 43.962105 256.793793 41.677831 218.808617

(3)^146/(2)^178/(5)^23 111.016515 1530.538820 95.204576 965.283848

(2)^92*(3)^191/(5)^170 148.123822 943.596569 154.481332 1070.384081

(2)^71*(5)^37/(3)^99 70.743669 181.089163 63.242667 129.377945

(5)^49/(2)^90/(3)^15 35.505868 34.075240 39.697565 47.624467

(5)^256/(2)^111/(3)^305 231.633331 1286.017248 238.618149 1405.898784

(3)^153*(5)^73/(2)^412 163.109370 226.729751 143.955865 155.869043

(2)^161/(3)^84/(5)^12 73.972968 48.546279 63.238794 30.331058

(2)^21*(3)^290/(5)^207 211.217108 235.136329 210.059812 231.292406

(2)^140*(5)^195/(3)^374 264.538592 419.006551 247.279087 342.228257

(3)^237*(5)^85/(2)^573 235.992938 74.673031 205.656081 49.418713

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How the harmonic entropy approach could be conciliate with this approach?

Goethe said: (excuse for that bad translation from French) "Do you want to penetrate the infinity? Go always
away and in all directions in the finity."

In that sense I understand well the necessity to go away and far in all directions at same time, but it could be
already useful to confront, relatively to our capacity to perceive a kind of harmonic order, the over optimism
implied by the fanciest commas, with the slight pessimism, in my opinion, of the harmonic entropy.

( It's not there an attack against the harmonic entropy. I believe only the approach appears slightly pessimistic
for the level is the sensation one, so neglecting contextual reinforcement arising at perception level. )

Once the word ridiculous has been used, as to suggest one have to think at the limit, one have to elaborate on
the sense of a such limit. There is not only one valid perspective, but I believe we have to distinguish minimally
the perceptual one concerning the music and the technical one concerning the luthery. Where abondant decimals
of cents may have sense in the technical perspective, it becomes rather ridiculous (indeed) in the musical
perspective.

So, I invite to reflexion in view to precise sense and perspective of these few notions illustrated with so much
buch of numbers.

Pierre

--- In tuning-math@y..., "Kees van Prooijen" <kees@d...> wrote:
> Hi Gene,
>
> Did you ever look at my list here:
> http://www.kees.cc/tuning/s235.html

I didn't know about it; I think I'll poke around on your site. Your comma factorization is missing powers of 2, by the way.

i don't know what to say to pierre, other than that some may wish the
theory to cover more ground, instead of less, than any practical
application would require -- why not? -- that way you'll never
fall "off the edge", and given more examples, you can more easily
ponder the mathematical relationships, a process which can sometimes
offer insight applicable even on the most "worldly" scale.

that said, none of these "ridiculous" lists included commas of clear
practical import, the blackwood 256/243 and the negri 16875/16384. so
gene did not answer my question.

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> that said, none of these "ridiculous" lists included commas of clear
> practical import, the blackwood 256/243 and the negri 16875/16384. so
> gene did not answer my question.

You asked for a list based on log-flat badness, and I gave you one.

> http://www.kees.cc/tuning/s235.html

> I didn't know about it; I think I'll poke around on your site. Your comma
factorization is missing powers of 2, by the way.

They're there now

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > that said, none of these "ridiculous" lists included commas of
clear
> > practical import, the blackwood 256/243 and the negri
16875/16384. so
> > gene did not answer my question.
>
> You asked for a list based on log-flat badness, and I gave you one.

i said "i'd like to know what is missing, based on some
log-flat badness measure." i didn't say "i'd like to know what to
eliminate, based on some log-flat badness measure." at least, there's
no way i'm eliminating negri and blackwood.

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> i said "i'd like to know what is missing, based on some
> log-flat badness measure." i didn't say "i'd like to know what to
> eliminate, based on some log-flat badness measure." at least, there's
> no way i'm eliminating negri and blackwood.

If we used weighted or geometric badness and upped the cutoff a little these would be included. You'd then need to explain what you mean by "missing"--obviously, at some point you need to close off the list!

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > i said "i'd like to know what is missing, based on some
> > log-flat badness measure." i didn't say "i'd like to know what to
> > eliminate, based on some log-flat badness measure." at least,
there's
> > no way i'm eliminating negri and blackwood.
>
> If we used weighted or geometric badness and upped the cutoff a
>little these would be included.

that's what i was trying to get you to provide.

>You'd then need to explain what you mean by "missing"

just the above -- no more, no less!

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> >You'd then need to explain what you mean by "missing"
>
> just the above -- no more, no less!

I guess /tuning-math/message/4554
is a step in the right direction, at least.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > >You'd then need to explain what you mean by "missing"
> >
> > just the above -- no more, no less!
>
> I guess /tuning-math/message/4554
> is a step in the right direction, at least.

sure -- i just wouldn't want to cut it off before the monzisma, out
of admiration for the Monz.

Regarding the link below. What are the vectors calculated from? Are they
wedge-invariants? Why are there two for each comma? Sorry if this seems
like a dumb question, but I'd like to be able to calculate these myself.
THANKS

wallyesterpaulrus
<wallyesterpaulrus To: tuning-math@yahoogroups.com
@yahoo.com> cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Even more ridiculous 5-comma list
11/26/2002 12:15
PM
Please respond to
tuning-math

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > >You'd then need to explain what you mean by "missing"
> >
> > just the above -- no more, no less!
>
> I guess /tuning-math/message/4554
> is a step in the right direction, at least.

sure -- i just wouldn't want to cut it off before the monzisma, out
of admiration for the Monz.

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--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Regarding the link below. What are the vectors calculated from? Are they
> wedge-invariants? Why are there two for each comma? Sorry if this seems
> like a dumb question, but I'd like to be able to calculate these myself.
> THANKS

The two vectors are actually to be thought of as two columns of a matrix; the matrix is the period/generator mapping, the first column giving how many periods, and the second how many generators, for the primes 2, 3 and 5. Wedging them together gives a wedge-invariant, but in the 5-limit case this can be identified with the comma. There are various ways to calculate such a matrix--for example, take two ets which define the temperament and reduce it to Hermite normal form.

Thanks for the explanation. I'll admit I am a little stuck on how there can
be negative numbers (for generators). I will continue to study this. Very
interesting....

"Gene Ward Smith"
<genewardsmith@ju To: tuning-math@yahoogroups.com
no.com> cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Even more ridiculous 5-comma list
11/26/2002 06:43
PM
Please respond to
tuning-math

--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Regarding the link below. What are the vectors calculated from? Are they
> wedge-invariants? Why are there two for each comma? Sorry if this seems
> like a dumb question, but I'd like to be able to calculate these myself.
> THANKS

The two vectors are actually to be thought of as two columns of a matrix;
the matrix is the period/generator mapping, the first column giving how
many periods, and the second how many generators, for the primes 2, 3 and
5. Wedging them together gives a wedge-invariant, but in the 5-limit case
this can be identified with the comma. There are various ways to calculate
such a matrix--for example, take two ets which define the temperament and
reduce it to Hermite normal form.

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--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...>
wrote:
>
> Thanks for the explanation. I'll admit I am a little stuck on how
there can
> be negative numbers (for generators).

can you give an example of where you're getting stuck? i'm sure we
can hold your hand and walk you through this . . .

Thanks. Actually I spoke too soon about the negative numbers. Upon further
examination they make perfect sense. I still don't understand the "period"
vector though. What is it measuring? (The "generator" vector makes sense.)
For example:

250/243 (2)*(5)^3/(3)^5 porcupine
[[1, 2, 3], [0, -3, -5]]

comp 5.948285733 rms 7.975800816 bad 1678.609846
generators [1200., 162.9960265]

What does [1,2,3] measure in terms of the primes 2,3 and 5?

THANKS

wallyesterpaulrus
<wallyesterpaulrus To: tuning-math@yahoogroups.com
@yahoo.com> cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Even more ridiculous 5-comma list
11/27/2002 01:28
PM
Please respond to
tuning-math

--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...>
wrote:
>
> Thanks for the explanation. I'll admit I am a little stuck on how
there can
> be negative numbers (for generators).

can you give an example of where you're getting stuck? i'm sure we
can hold your hand and walk you through this . . .

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--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> 250/243 (2)*(5)^3/(3)^5 porcupine

[[1, 2, 3]
[0,-3,-5]]

> comp 5.948285733 rms 7.975800816 bad 1678.609846
> generators [1200.,162.9960265]

1*1200 - 0*163 = 1200 (2)

2*1200 - 3*163 = 1911 (3)

3*1200 - 5*163 = 2785 (5)

--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...>
wrote:
>
> Thanks. Actually I spoke too soon about the negative numbers. Upon
further
> examination they make perfect sense. I still don't understand
the "period"
> vector though. What is it measuring? (The "generator" vector makes
sense.)
> For example:
>

>

>

> 250/243 (2)*(5)^3/(3)^5
porcupine
> [[1, 2, 3], [0, -3, -
5]]
>

>

>

>

> comp 5.948285733 rms 7.975800816 bad
1678.609846
> generators [1200.,
162.9960265]
>

>

>

>

>
> What does [1,2,3] measure in terms of the primes 2,3 and 5?

you need both columns together. the first column (this one) refers to
the first generator, or period. the second column ([0 -3 -5]) refers
to the second generator, usually referred to simply as *the*
generator.

so in this case,

prime 2 is approximated by 1*1200 + 0*162.9960265

prime 3 is approximated by 2*1200 + -3*162.9960265

prime 5 is approximated by 3*1200 + -5*162.9960265

Thanks. Now just one more question: How do you calculate the second
generator (e.g. 162.9960265 in "porcupine")? I get slightly different
values, ones that still work, but I am wondering why they are different
from your's. Makes me wonder if I am simplifying something incorrectly. For
example, in "kleismic" I just use ln(6/5)/ln(2) * 1200, which is 315.641287
instead of 317.079753.

Paul

"Gene Ward Smith"
<genewardsmith@ju To: tuning-math@yahoogroups.com
no.com> cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Even more ridiculous 5-comma list
11/27/2002 06:43
PM
Please respond to
tuning-math

--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> 250/243 (2)*(5)^3/(3)^5 porcupine

[[1, 2, 3]
[0,-3,-5]]

> comp 5.948285733 rms 7.975800816 bad 1678.609846

> generators [1200.,162.9960265]

1*1200 - 0*163 = 1200 (2)

2*1200 - 3*163 = 1911 (3)

3*1200 - 5*163 = 2785 (5)

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--- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...>
wrote:
>
> Thanks. Now just one more question: How do you calculate the second
> generator (e.g. 162.9960265 in "porcupine")? I get slightly
different
> values, ones that still work, but I am wondering why they are
different
> from your's. Makes me wonder if I am simplifying something
incorrectly. For
> example, in "kleismic" I just use ln(6/5)/ln(2) * 1200, which is
315.641287
> instead of 317.079753.
>
> Paul

looks like you're just using the pure minor third for kleismic. maybe
you're using minimax optimization instead of rms?

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> --- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...>
> wrote:
> >
> > Thanks. Now just one more question: How do you calculate the
second
> > generator (e.g. 162.9960265 in "porcupine")? I get slightly
> different
> > values, ones that still work, but I am wondering why they are
> different
> > from your's. Makes me wonder if I am simplifying something
> incorrectly. For
> > example, in "kleismic" I just use ln(6/5)/ln(2) * 1200, which is
> 315.641287
> > instead of 317.079753.
> >
> > Paul
>
> looks like you're just using the pure minor third for kleismic.
maybe
> you're using minimax optimization instead of rms?
How would one use rms to get 317.079753?
Paul

--- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...> wrote:

> How would one use rms to get 317.079753?
> Paul

for 5-limit temperaments, you minimize the rms error in the 3/1, the
5/1, and the 5/3 -- in other words, all three 5-limit consonant
interval classes.

-another paul

Thanks. I'm starting to get it. Could you show me the actual calculations
for the second generator (rms error) for say, meantone?

wallyesterpaulrus
<wallyesterpaulrus To: tuning-math@yahoogroups.com
@yahoo.com> cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
Subject: [tuning-math] Re: Even more ridiculous 5-comma list
11/30/2002 01:49
PM
Please respond to
tuning-math

--- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...> wrote:

> How would one use rms to get 317.079753?
> Paul

for 5-limit temperaments, you minimize the rms error in the 3/1, the
5/1, and the 5/3 -- in other words, all three 5-limit consonant
interval classes.

-another paul

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hi paul,

for meantone, see this page:

http://sonic-arts.org/monzo/woolhouse/essay.htm

-paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Thanks. I'm starting to get it. Could you show me the actual
calculations
> for the second generator (rms error) for say, meantone?
>
>
>

>
wallyesterpaulrus

> <wallyesterpaulrus To: tuning-
math@yahoogroups.com
> @yahoo.com> cc: (bcc:
Paul G Hjelmstad/US/AMERICAS)
> Subject: [tuning-
math] Re: Even more ridiculous 5-comma list
> 11/30/2002
01:49

>
PM

> Please respond
to

> tuning-
math

>

>

>
>
>
>
> --- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...>
wrote:
>
> > How would one use rms to get 317.079753?
> > Paul
>
> for 5-limit temperaments, you minimize the rms error in the 3/1, the
> 5/1, and the 5/3 -- in other words, all three 5-limit consonant
> interval classes.
>
> -another paul
>
>
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Thanks. Worked through Woolhouse/Monzo's derivation of 7/26-meantone. Now I
see how the generators are calculated. Now, how are (the other) rms,
complexity and badness calculated?

Paul

"wallyesterpaulrus
<wallyesterpaulrus@ To: tuning-math@yahoogroups.com
yahoo.com>" cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
<wallyesterpaulrus Subject: [tuning-math] Re: Even more ridiculous 5-comma list

12/09/2002 04:56 PM
Please respond to
tuning-math

hi paul,

for meantone, see this page:

http://sonic-arts.org/monzo/woolhouse/essay.htm

-paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Thanks. I'm starting to get it. Could you show me the actual
calculations
> for the second generator (rms error) for say, meantone?
>
>
>

>
wallyesterpaulrus

> <wallyesterpaulrus To: tuning-
math@yahoogroups.com
> @yahoo.com> cc: (bcc:
Paul G Hjelmstad/US/AMERICAS)
> Subject: [tuning-
math] Re: Even more ridiculous 5-comma list
> 11/30/2002
01:49

>
PM

> Please respond
to

> tuning-
math

>

>

>
>
>
>
> --- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...>
wrote:
>
> > How would one use rms to get 317.079753?
> > Paul
>
> for 5-limit temperaments, you minimize the rms error in the 3/1, the
> 5/1, and the 5/3 -- in other words, all three 5-limit consonant
> interval classes.
>
> -another paul
>
>
> Yahoo! Groups Sponsor
>
>
>
> ADVERTISEMENT
> (Embedded image moved to file: pic04186.gif)
>
>
> (Embedded image moved to file: pic19690.gif)
>
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

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

Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

rms error is the whole focus of the woolhouse calculation. it
minimizes the rms error. complexity is a function of the mapping, and
badness is a function of both complexity and rms error.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Thanks. Worked through Woolhouse/Monzo's derivation of 7/26-
meantone. Now I
> see how the generators are calculated. Now, how are (the other) rms,
> complexity and badness calculated?
>
> Paul
>
>
>

> "wallyesterpaulrus

> <wallyesterpaulrus@ To: tuning-
math@yahoogroups.com
> yahoo.com>" cc: (bcc:
Paul G Hjelmstad/US/AMERICAS)
> <wallyesterpaulrus Subject: [tuning-
math] Re: Even more ridiculous 5-comma list
>

> 12/09/2002 04:56
PM

> Please respond
to

> tuning-
math

>

>

>
>
>
>
> hi paul,
>
> for meantone, see this page:
>
> http://sonic-arts.org/monzo/woolhouse/essay.htm
>
> -paul
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> >
> > Thanks. I'm starting to get it. Could you show me the actual
> calculations
> > for the second generator (rms error) for say, meantone?
> >
> >
> >
>
> >
> wallyesterpaulrus
>
> > <wallyesterpaulrus To: tuning-
> math@yahoogroups.com
> > @yahoo.com> cc: (bcc:
> Paul G Hjelmstad/US/AMERICAS)
> > Subject: [tuning-
> math] Re: Even more ridiculous 5-comma list
> > 11/30/2002
> 01:49
>
> >
> PM
>
> > Please respond
> to
>
> > tuning-
> math
>
> >
>
> >
>
> >
> >
> >
> >
> > --- In tuning-math@y..., "paulhjelmstad" <paul.hjelmstad@u...>
> wrote:
> >
> > > How would one use rms to get 317.079753?
> > > Paul
> >
> > for 5-limit temperaments, you minimize the rms error in the 3/1,
the
> > 5/1, and the 5/3 -- in other words, all three 5-limit consonant
> > interval classes.
> >
> > -another paul
> >
> >
> > Yahoo! Groups Sponsor
> >
> >
> >
> > ADVERTISEMENT
> > (Embedded image moved to file: pic04186.gif)
> >
> >
> > (Embedded image moved to file: pic19690.gif)
> >
> >
> >
> >
> > To unsubscribe from this group, send an email to:
> > tuning-math-unsubscribe@yahoogroups.com
> >
> >
> >
> > Your use of Yahoo! Groups is subject to the Yahoo! Terms of
Service.
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.