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Message: 8900 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 22:14:54

Subject: Ragmonza 225/224 planar

From: Gene Ward Smith

When tuning 225/224-planar (which used to be called both Byzantine and
Pauline until people objected) there is no payoff in not having minor
thirds pure (the approximation to 7 involving 3 and 5 only in their
product of 15) and optimizing makes it pure. If we assume pure 6/5s
(and octaves, of course) we get a one-parameter family of
225/224-planar temperaments, rather as if it were a linear temperament.
If s is the amount by which the thirds and fifths are flat, we have

q3 = 3/s

q5 = 5/s

q7 = 225/(32 s^4)

for the size of the "3", "5", and "7". If s = 32805/32768 is exactly a
schisma, we get a 5-limit version of 225/224-planar

r3 = 32768/10935

r5 = 32768/6561

r7 = |55 -30 2>

r7 is shy of 7 by a comma which is ragisma/monzisma, which I guess we
may call the ragmonza of |-55 30 2 1>.

Another plausible 225/224 planar tuning sets the 7s to be pure, which
makes s = (225/224)^(1/4), a schisma of 1.928 versus one of 1.954. We
may also want to make 7/5 and 7/6 pure, which means 
s = (225/224)^(1/3) and we flatten thirds and fifths by 2.571 cents.

Message: 8901 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 22:20:19

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 

>> No, Pajara is simply the 7-limit extension of Diaschsimic that 
you do 
>> get in 22-equal (and pretty much in no other ET):
>> 
>> GX Networks * [with cont.]  (Wayb.)
>> 
>> As long as you're talking 5-limit though, there's no reason to 
bring 
>> Pajara into it.
> 

> We are tuning Diaschismic so that the 7-limit works well, whether we
> want to acknowledge that fact or not.


You mean in the particular case of 22-equal?


> It's a Pajara tuning of
> Diaschismic, in other words, and so correctly called Pajara.


By "It" you mean 22-equal?


> Your method gives two precisely equal scales, one of which is called
> Diaschismic[12] and the other Pajara[12].

My method?

Message: 8902 - Contents - Hide Contents

Date: Thu, 01 Jan 2004 23:20:39

Subject: Re: The Two Diadie Scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


>> We are tuning Diaschismic so that the 7-limit works well, whether 
we
>> want to acknowledge that fact or not.
> 

> You mean in the particular case of 22-equal?


Or any other tuning where the fifth is quite sharp. You'd have us 
call it "Diaschismic" even if the tuning was the pure 9/7's tuning!


>> Your method gives two precisely equal scales, one of which is 
called
>> Diaschismic[12] and the other Pajara[12].
> 
> My method?


Your scale naming method.

Message: 8903 - Contents - Hide Contents

Date: Fri, 02 Jan 2004 18:07:32

Subject: 9-limit

From: Carl Lumma

By the way, Gene, you never got back to me about 9-limit tunings.
For any 7-limit scale, can't we produce a 9-limit tuning simply
by using a multiplier of 3 on the 3:2 error before optimizing?

And when you give, for example, 11-limit Marvelous Class, are
you counting the 3:2 error once or thrice?

-Carl

Message: 8904 - Contents - Hide Contents

Date: Fri, 02 Jan 2004 18:15:54

Subject: Re: 9-limit

From: Carl Lumma

>> >or any 7-limit scale, can't we produce a 9-limit tuning simply
>> by using a multiplier of 3 on the 3:2 error before optimizing?
>

>No. I don't know why you would use a multiplier of 3, and you'd 
>obviously be ignoring 9:5, 9:7, and 9:8. Unless you're assuming a 
>particular error function, but still don't think it could work.


Sorry, you're right of course.

-C.

Message: 8905 - Contents - Hide Contents

Date: Fri, 02 Jan 2004 00:01:40

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>>> We are tuning Diaschismic so that the 7-limit works well, 
whether 
> we

>>> want to acknowledge that fact or not.
>> 

>> You mean in the particular case of 22-equal?
> 

> Or any other tuning where the fifth is quite sharp. You'd have us 
> call it "Diaschismic" even if the tuning was the pure 9/7's tuning!


But this is nonsense. Why would you use 22-equal or any good Pajara 
tuning when there are far better Diaschismic tunings for doing what 
you mention in connection with Pajara in all your recent posts? All 
those posts concerned 5-limit only. What if we were to substitute 
the "Dominant Sevenths" temperament. for "Meantone" in all those 
posts? Would you be happy?


>>> Your method gives two precisely equal scales, one of which is 
> called

>>> Diaschismic[12] and the other Pajara[12].
>> 
>> My method?
> 

> Your scale naming method.


Not any more than yours would give two precisely equal scales, one of 
which is called DominantSevenths[12] and the other Meantone[12].

Message: 8906 - Contents - Hide Contents

Date: Fri, 02 Jan 2004 05:57:22

Subject: Re: The Two Diadie Scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


>> Or any other tuning where the fifth is quite sharp. You'd have us 
>> call it "Diaschismic" even if the tuning was the pure 9/7's 
tuning!
> 

> But this is nonsense. Why would you use 22-equal or any good Pajara 
> tuning when there are far better Diaschismic tunings for doing what 
> you mention in connection with Pajara in all your recent posts? 


Because I want higher than 5-limit harmonies, of course.

All 

> those posts concerned 5-limit only. 


Not if you read carefully. I mentioned the various approximate higher 
limit consonances, after all.


> Not any more than yours would give two precisely equal scales, one 
of 
> which is called DominantSevenths[12] and the other Meantone[12].


I'd call 12-equal both of these, but I'd hardly term the 31-et 
version of DominantSevenths[12] by that name.

Message: 8907 - Contents - Hide Contents

Date: Fri, 02 Jan 2004 07:07:54

Subject: The Three Majraj Scales

From: Gene Ward Smith

The classification effort staggers on.

! majraj1.scl
Majraj 648/625 6561/6250 scale
12
!
27/25
125/108
729/625
5/4
27/20
25/18
3/2
81/50
5/3
9/5
243/125
2

! majraj2.scl
Majraj 648/625 6561/6250 scale
12
!
27/25
10/9
6/5
162/125
4/3
36/25
3/2
125/81
5/3
9/5
50/27
2

! majraj3.scl
Majraj 648/625 6561/6250 scale
12
!
27/25
10/9
6/5
162/125
4/3
25/18
3/2
125/81
5/3
9/5
50/27
2

Message: 8908 - Contents - Hide Contents

Date: Fri, 02 Jan 2004 21:08:27

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>>> Or any other tuning where the fifth is quite sharp. You'd have 
us 
>>> call it "Diaschismic" even if the tuning was the pure 9/7's 
> tuning!
>> 

>> But this is nonsense. Why would you use 22-equal or any good 
Pajara 
>> tuning when there are far better Diaschismic tunings for doing 
what 
>> you mention in connection with Pajara in all your recent posts? 
> 

> Because I want higher than 5-limit harmonies, of course.
> 
> All 

>> those posts concerned 5-limit only. 
> 

> Not if you read carefully.


I do read carefully. Actually, my brain is beginning to deteriorate, 
I should join a gym.


> I mentioned the various approximate higher 
> limit consonances, after all.


I saw no mention of those in connection with the "Pajara" cases.



________________________________________________________________________
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Message: 8909 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 00:37:53

Subject: Re: The Two Diadie Scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


>> I mentioned the various approximate higher 
>> limit consonances, after all.
> 

> I saw no mention of those in connection with the "Pajara" cases.


It was in the same posting as the rest of them. What did you think I 
meant by saying it was a good candidate for 22-et tempering?

Message: 8910 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 20:51:20

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

From: Paul Erlich

Looks like this approach optimizes according to (Tenney) weighted 
minimax over ALL intervals.

IS THIS RIGHT??

No need to specify a consonance limit? -- wow that's hot.

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> No one commented on the graphs I posted around Christmas, but I'll 
> keep going, if only for myself . . .
> 
> The syntonic comma = 81/80 (21.506 cents) has a length of
> log(81*80) = 4*log(2) + 4*log(3) + log(5)
> in the Tenney lattice, as it comprises 4 rungs along the 2-axis, 4 
> rungs along the 3-axis, and 1 rung along the 5-axis.
> 
> If the anomaly is distributed uniformly and efficiently along this 
> length, the rungs along the 2-axis and 5-axis will be tempered 
wide, 
> and the rung along the 3-axis will be tempered narrow.
> 
> So the 2-axis rungs should be tempered wide by log(2)/log(81*80) of 
> the syntonic comma, or 1.6985 cents -- so each rung represents 
> 1201.6985 cents. The 3-axis rungs should be tempered narrow by log
> (3)/log(81*80) of the syntonic comma, or 2.6921 cents -- so each 
rung 
> represents 1899.2629 cents. The 5-axis rungs should be tempered 
wide 
> by log(5)/log(81*80) of the syntonic comma, or 3.9438 cents -- so 
> each rung represents 2790.2575 cents.
> 
> Check that the syntonic comma vanishes:
> 
> 4*1899.2629 - 4*1201.6985 - 2790.2575 = 0
> 
> What would you call this kind of meantone? The 3:2 is flattened by 
> about 10/49-comma, while the octave is widened by about 3/38-comma.
> 
> Looks like this approach optimizes according to a weighted minimax 
> over the prime intervals. Comments?

Message: 8911 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 00:49:40

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>>> I mentioned the various approximate higher 
>>> limit consonances, after all.
>> 

>> I saw no mention of those in connection with the "Pajara" cases.
> 

> It was in the same posting as the rest of them.


I've searched and found tuning post #50759, and tuning-math post 
#8332, and in both of them Pajara seems like a leap since you make no 
mention of anything beyond 5-limit. You start with Meantone[12], so 
if anything, the logical comparison would be to Diaschismic[12].

Another thing -- perhaps some of the non-Pajara 7-limit extensions of 
Diaschsimic would in fact be useful for some of those scales?

Message: 8912 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 20:57:30

Subject: Re: name?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> Is there a name for the 5-limit interval [-30 13], or the
> temperament which removes it?
> 
> -Carl


I doubt it. The interval in question is either

160000000000000
--------------- = -436.57 cents
205891132094649

or

320000000000000
--------------- = 763.43 cents
205891132094649

. . .

Message: 8913 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 01:39:01

Subject: Re: The Two Diadie Scales

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> I've searched and found tuning post #50759, and tuning-math post 
> #8332, and in both of them Pajara seems like a leap since you make 
no 
> mention of anything beyond 5-limit. You start with Meantone[12], so 
> if anything, the logical comparison would be to Diaschismic[12].


I hope you don't assume that I always mean 5-limit when I 
say "meantone". :)


> Another thing -- perhaps some of the non-Pajara 7-limit extensions 
of 
> Diaschsimic would in fact be useful for some of those scales?


I'm planning to see how many distinct scales one obtains in various 
temperaments after completing the classification.

Message: 8914 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 13:39:49

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

From: Carl Lumma

>> >o one commented on the graphs I posted around Christmas, but I'll 
>> keep going, if only for myself . . .


Gene and you had an exchange.  Gene suggested what I was thinking,
you said that each band represented only a denominator, not a comma.
Or are you talking about something else?


>No need to specify a consonance limit? -- wow that's hot.


How do you mean?  Isn't this implicit in the dimensionality of the
Tenney lattice you're using?

It would indeed be hot.  I wish I had a better grasp of Gene's zeta
stuff.

-Carl

Message: 8915 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 01:57:39

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>> I've searched and found tuning post #50759, and tuning-math post 
>> #8332, and in both of them Pajara seems like a leap since you 
make 
> no 

>> mention of anything beyond 5-limit. You start with Meantone[12], 
so 
>> if anything, the logical comparison would be to Diaschismic[12].
> 

> I hope you don't assume that I always mean 5-limit when I 
> say "meantone". :)


Since the PBs were 5-limit to begin with, it was by far the most 
natural assumption in this case.

Message: 8916 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 13:42:36

Subject: Re: name?

From: Carl Lumma

>> >s there a name for the 5-limit interval [-30 13], or the
>> temperament which removes it?
>> 
>> -Carl
>

>I doubt it. The interval in question is either
>
>160000000000000
>--------------- = -436.57 cents
>205891132094649
>
>or
>
>320000000000000
>--------------- = 763.43 cents
>205891132094649
>
>. . .


That's funny, it's supposed to be 222 cents.  Crap, so sorry,
it should have been [-30 0 13].

-Carl

Message: 8917 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 23:07:57

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>>> No one commented on the graphs I posted around Christmas, but 
I'll 
>>> keep going, if only for myself . . .
> 

> Gene and you had an exchange.  Gene suggested what I was thinking,
> you said that each band represented only a denominator, not a comma.
> Or are you talking about something else?


Yes, a later pair of graphs.


>> No need to specify a consonance limit? -- wow that's hot.
> 

> How do you mean?  Isn't this implicit in the dimensionality of the
> Tenney lattice you're using?


No, no consonance limit (aka odd-limit) is implicity in the 
dimensionality of the Tenney lattice you're using -- and even the 
latter, aka prime-limit, doesn't need to be specified -- for example 
the results for the pythagorean comma will be valid in both lower and 
higher prime limits.

Message: 8918 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 02:13:23

Subject: Re: 9-limit

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> For any 7-limit scale, can't we produce a 9-limit tuning simply
> by using a multiplier of 3 on the 3:2 error before optimizing?


No. I don't know why you would use a multiplier of 3, and you'd 
obviously be ignoring 9:5, 9:7, and 9:8. Unless you're assuming a 
particular error function, but still don't think it could work.

Message: 8919 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 23:09:39

Subject: Re: name?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>>> Is there a name for the 5-limit interval [-30 13], or the
>>> temperament which removes it?
>>> 
>>> -Carl
>> 

>> I doubt it. The interval in question is either
>> 
>> 160000000000000
>> --------------- = -436.57 cents
>> 205891132094649
>> 
>> or
>> 
>> 320000000000000
>> --------------- = 763.43 cents
>> 205891132094649
>> 
>> . . .
> 

> That's funny, it's supposed to be 222 cents.


Well, then I still doubt it.


> Crap, so sorry,
> it should have been [-30 0 13].


Oh so you're thinking about 13-equal.

Message: 8920 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 15:19:01

Subject: Re: name?

From: Carl Lumma

>Oh so you're thinking about 13-equal.


Yup.  And I think only multiples of 13 do the job?

-Carl

Message: 8921 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 03:54:39

Subject: non-1200: Tenney/heursitic meantone temperament

From: Paul Erlich

No one commented on the graphs I posted around Christmas, but I'll 
keep going, if only for myself . . .

The syntonic comma = 81/80 (21.506 cents) has a length of
log(81*80) = 4*log(2) + 4*log(3) + log(5)
in the Tenney lattice, as it comprises 4 rungs along the 2-axis, 4 
rungs along the 3-axis, and 1 rung along the 5-axis.

If the anomaly is distributed uniformly and efficiently along this 
length, the rungs along the 2-axis and 5-axis will be tempered wide, 
and the rung along the 3-axis will be tempered narrow.

So the 2-axis rungs should be tempered wide by log(2)/log(81*80) of 
the syntonic comma, or 1.6985 cents -- so each rung represents 
1201.6985 cents. The 3-axis rungs should be tempered narrow by log
(3)/log(81*80) of the syntonic comma, or 2.6921 cents -- so each rung 
represents 1899.2629 cents. The 5-axis rungs should be tempered wide 
by log(5)/log(81*80) of the syntonic comma, or 3.9438 cents -- so 
each rung represents 2790.2575 cents.

Check that the syntonic comma vanishes:

4*1899.2629 - 4*1201.6985 - 2790.2575 = 0

What would you call this kind of meantone? The 3:2 is flattened by 
about 10/49-comma, while the octave is widened by about 3/38-comma.

Looks like this approach optimizes according to a weighted minimax 
over the prime intervals. Comments?

Message: 8922 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 15:27:35

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

From: Carl Lumma

>> >ene and you had an exchange.  Gene suggested what I was thinking,
>> you said that each band represented only a denominator, not a comma.
>> Or are you talking about something else?
>

>Yes, a later pair of graphs.


D'oh!  Link?  (I even went to tuning-math to find it myself, but
this thread is not connected to anything).


>>> No need to specify a consonance limit? -- wow that's hot.
>> 

>> How do you mean?  Isn't this implicit in the dimensionality of the
>> Tenney lattice you're using?
>

>No, no consonance limit (aka odd-limit) is implicity in the 
>dimensionality of the Tenney lattice you're using -- and even the 
>latter, aka prime-limit, doesn't need to be specified -- for example 
>the results for the pythagorean comma will be valid in both lower and 
>higher prime limits.


I must not be tracking you -- the pythagorean comma is of course
a 3-limit comma.

-Carl

Message: 8923 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 04:57:17

Subject: Taxicab/Tenney version of heron's formula?

From: Paul Erlich

In order to quantify straightness, I tried to come up with a formula 
that would give the 'area' of a bivector regardless of which basis 
vectors are chosen to define it. For example, 12-tone periodicity 
blocks obviously all have the same 'area' regardless of whether 
they're defined using {81:80, 125:128}, {81:80, 648:625}, or what 
have you. But when dealing with, say, 7-limit linear temperaments, 
you can't simply quantify the area of the vanishing bivector using a 
number of notes. I tried heron's formula using the Tenney lengths of 
various equivalent dependent triples to determine the area (really 
half the area) of the bivector, and it's almost independent of which 
triple I use, but not quite. Any suggestions?

Message: 8924 - Contents - Hide Contents

Date: Sat, 03 Jan 2004 04:35:17

Subject: name?

From: Carl Lumma

Is there a name for the 5-limit interval [-30 13], or the
temperament which removes it?

-Carl



________________________________________________________________________
________________________________________________________________________




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