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Message: 8950 - Contents - Hide Contents

Date: Mon, 05 Jan 2004 15:52:13

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

From: Carl Lumma

>> >erhaps it has something to do with using this to get
>> optimum generators for a linear temperament?
>

>Well, that's exactly what this does (when the dimensionality is 
>right), as I've illustrated already in a few cases.
>
>Here's something new -- Top meantone is, it seems, exactly 1/4-comma 
>meantone (I get 0.24999999999997, but that's probably just rounding 
>error) except a uniform (in cents, or log Hz) stretch of 
>1.00141543374547 is applied to all intervals . . .


Is the formula for that particularly hard?


>> And I don't understand your 'limitless' claim -- since p/q contains
>> the factors it does and no others, one wouldn't expect its vanishing
>> to effect
>
>affect?


Yes, I think so...  :)


>> intervals different factors.
>

>intervals with different factors? Well, 5:4 and 5:3 have *some* 
>factors differing from those in the Pythagorean comma, yet both 
>intervals are affected by its vanishing, in this scheme.


But not 7:5, right?

-Carl

Message: 8951 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 19:31:00

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

From: Carl Lumma

>>> >ow, for all primes r,
>>> 
>>> If p contains any factors of r, the r-rungs in the lattice (which 
>>> have length log2(r)) are shrunk from
>>> cents(r)
>>> to
>>> cents(r) - log2(r)*cents(p/q)/log2(p*q).
>>> If q contains any factors of 2, they are instead stretched to
>>> cents(r) + log2(r)*cents(p/q)/log2(p*q).
>> 

>> Thanks.  I understand this 100%.  But I don't understand what's
>> new.
>

>Where have you seen this before?


I guess in my head.


>> Perhaps it has something to do with using this to get
>> optimum generators for a linear temperament?
>

>Well, that's exactly what this does (when the dimensionality is 
>right), as I've illustrated already in a few cases.


What if the generator isn't a just interval?  Then isn't it still
the same kind of multivariable optimization that you guys have been
using all along?


>> And I don't understand your 'limitless' claim -- since p/q contains
>> the factors it does and no others, one wouldn't expect its vanishing
>> to effect


What I mean is, when extending meantone to a 7-limit mapping, it
will naturally implicate different commas and change the optimal
generator a bit, same as before.  Well, we can't be sure until we
see how to combine commas.  But to claim it doesn't require a limit
when it's currently limited to linear temperaments in the 5-limit...
Or am I all wet?

-Carl

Message: 8952 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 19:41:12

Subject: Re: Meantone reduced blocks

From: Paul Erlich

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

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

>>> meatone reduction therefore is
>>> 
>>> 7i - 12(round(i/3) + round(i/4+1/8))
>> 

>> Please express this meantone scale in conventional letter-name-
and-
>> accidental notation.
> 

> Here is the scale in terms of meantone fifths:
> 
> [0, 7, -10, -3, 4, -1, -6, 1, -4, 3, -2, -7]
> 
> Using an "f" for my flat symbol, here it is in sharps and flats:
> 
> [C, C#, Eff, Ef, E, F, Gf, G, Af, A, Bf, Cf]
> 
> 
> I'm not sure yet how uncommon this sort of thing is, but probably 
not
> very common. There is only one possible {128/125, 648/625} scale up 
to
> transposition, and one question is what other comma pairs give us
> something differing from Meantone[12] on reduction.


Have you found any that don't reduce to something other than:

Meantone[12]
Diaschismic[12]
Augmented[12]
Diminished[12]

when tempered accordingly?

Message: 8953 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 21:28:05

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

From: Graham Breed

Me:

>> I thought this was all assumed by your hypothesis anyway.

Paul:

> I don't see the relationship.

From


   error = comma size / complexity

using Tenny complexity, for a comma n:d:

error = log(n/d) / log(n*d)
       = log[1 + (n-d)/d] / log[(n/d)*d*d]

The base of the logarithms doesn't matter, so we can use log(1+x) ~ x as 
(n-d)/d will be small for a comma.  so this is much the same as

error = [(n-d)/d]/[log(d*d) + log(n/d)]
       = (n-d)/d/2/log(d) + 1/d

as it happens.  The 1/d term is small and can be neglected, giving

error = (n-d)/[2d*log(d)]

The heuristic error given here:

Definitions of tuning terms: heuristic error, ... * [with cont.]  (Wayb.)

is |n-d|/(R*log(R))

for commas as usually written, the numerator is larger than the 
denominator, so the |n-d| is the same as (n-d).  Depending on whether n 
or d is even, R may be the same as D.  If it isn't exactly the same, 
it's going to be close because commas tend to be small (and have to be 
for the simplification to work) which means the numerator and 
denominator are of roughly equal size.  The factor of 2 is disposable, 
as this isn't measuring anything in particular.

So they look pretty similar to me.


> Yes, but for octave equivalence (pegged to 1200 cent octaves), I'd 
> like to eventually be able to use Kees's expressibility measure 
> instead of Tenney harmonic distance. Just as there was no 
> finitistic 'limit' assumed for my 'optimization' in the Tenney 
> lattice, no odd limit will have to be specified in the octave-
> equivalent case (if it can work).


I don't see why it shouldn't work mathematically.  Whether it has any 
musical meaning is a different matter.  But why shouldn't it work?  You 
temper each factor in whichever of n and d is odd, or the larger if they 
both are.


>> As geometric complexity looks like 
>> being an octave-specific weighted complexity measure, this may be 
> the 

>> way to progress.
> 

> What do you mean?


Odd limits are a simplification so that we always get whole numbers, and 
can think octave equivalently.  The geometric complexity Gene gave, as 
far as I could understand it, was naturally continuous and octave 
specific.  So if it makes it easier to work that way, we can, and go 
back to odd limits for the fine tuning.



>> The problem remains knowing how best to combine these commas to get 
> a 

>> temperament of a specific dimension.  For that we need a 
> straightness 

>> measure, as always.
> 
> 

> That's why I was asking about heron's formula, etc. But if we have 
> some way of acheiving this Tenney-weighted minimax for the relevant 
> temperaments, we may be able to skip this step.


I don't see how we can skip the step of combining commas.  How could it 
make sense to do so?


                     Graham

Message: 8954 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 21:46:47

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Me:

>>> I thought this was all assumed by your hypothesis anyway.
> 
> Paul:

>> I don't see the relationship.
> 
> From
> 

>    error = comma size / complexity
> 
> using Tenny complexity, for a comma n:d:
> 
> error = log(n/d) / log(n*d)
>        = log[1 + (n-d)/d] / log[(n/d)*d*d]


Oh, you mean "heuristic", not "hypothesis" (the latter concerns PBs 
and DEs).


>> Yes, but for octave equivalence (pegged to 1200 cent octaves), 
I'd 
>> like to eventually be able to use Kees's expressibility measure 
>> instead of Tenney harmonic distance. Just as there was no 
>> finitistic 'limit' assumed for my 'optimization' in the Tenney 
>> lattice, no odd limit will have to be specified in the octave-
>> equivalent case (if it can work).
> 

> I don't see why it shouldn't work mathematically.  Whether it has 
any 
> musical meaning is a different matter.  But why shouldn't it work?  
You 
> temper each factor in whichever of n and d is odd, or the larger if 
they 
> both are.


Hmm . . . I'm a bit busy right now, so can you work out an example, 
and show all the errors like I did with Top meantone here?


>>> As geometric complexity looks like 
>>> being an octave-specific weighted complexity measure, this may be 
>> the 

>>> way to progress.
>> 

>> What do you mean?
> 

> Odd limits are a simplification so that we always get whole 
numbers, and 
> can think octave equivalently.  The geometric complexity Gene gave, 
as 
> far as I could understand it, was naturally continuous and octave 
> specific.  So if it makes it easier to work that way, we can, and 
go 
> back to odd limits for the fine tuning.


I'll have to put this aside for later digestion . . .


>>> The problem remains knowing how best to combine these commas to 
get 
>> a 

>>> temperament of a specific dimension.  For that we need a 
>> straightness 

>>> measure, as always.
>> 
>> 

>> That's why I was asking about heron's formula, etc. But if we 
have 
>> some way of acheiving this Tenney-weighted minimax for the 
relevant 
>> temperaments, we may be able to skip this step.
> 

> I don't see how we can skip the step of combining commas.  How 
could it 
> make sense to do so?


I meant skip the step of getting a straightness measure. If we get 
the right straightness measure, it won't matter which kernel basis we 
pick for a given temperament. In which case we may be able to proceed 
directly from the kernel to the relevant quantities. But I wouldn't 
stress it at the moment . . .

Thanks for being you!

Message: 8955 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 22:35:26

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

From: Graham Breed

Paul Erlich wrote:

> Maybe someone can derive this 0.24999999999997 as a 1/4 symbolically. 
> I'd be very happy to see it.


1.0014154337454717 you say?  Yes, I've derived it symbolically -- which 
means I must have duplicated your method for tempering.  That magic 
number is 2*log(81)/log(81*80).  The calculation goes something this:

The 2 generator is tempered as cents(2)[1+k]

The 3 generator is tempered as cents(3)[1-k]

The 5 generator is tempered as cents(5)[1+k]

where k is log(81/80)/log(81*80).  The + or - depends on whether factors 
of this prime occur in the denominator or numerator respectively.

So as 2 and 5 are just in quarter comma meantone, they must be stretched 
by 1+k here.

In quarter comma meantone, the 3 generator is tempered as

cents(3) - cents(81/80)/4

= cents(3) - cents(81)/4 + cents(80)/4
= cents(3) - cents(3) + cents(80)/4
= cents(80)/4

so the stretch from that to its new tempered value is

cents(3)[1-k]/[cents(80)/4]

= 4*cents(3)([1-k]/cents(80)
= [1-k]cents(81)/cents(80)

Now we need to substitute in k, so that stretch becomes

[1-cents(81/80)/cents(81*80)]*cents(81)/cents(80)

(cents are a special case of log)

= [(cents(81*80) - cents(81/80))/cents(81*80)]*cents(81)/cents(80)
= [2*cents(80)/cents(81*80)]*cents(81)/cents(80)
= 2*cents(81)/cents(81*80)
= 2*log(81)/log(81*80)

which is the same as 1+k, and you can work through that if you like.


                  Graham

Message: 8956 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 22:44:49

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

From: Graham Breed

Paul Erlich wrote:


> Oh, you mean "heuristic", not "hypothesis" (the latter concerns PBs 
> and DEs).


Yes, well, I typed the right thing into Google anyway :-)


> Hmm . . . I'm a bit busy right now, so can you work out an example, 
> and show all the errors like I did with Top meantone here?


Meantone's the easy one.  81 is the odd part of 81/80.  So we need to 
share the error of 81/80 amongst the factors of 81 according to their 
respective weights.  As 3 is the only prime factor of 81, it takes all 
the tempering, and each 3 gets a quarter of it, hence quarter comma 
meantone.


> I meant skip the step of getting a straightness measure. If we get 
> the right straightness measure, it won't matter which kernel basis we 
> pick for a given temperament. In which case we may be able to proceed 
> directly from the kernel to the relevant quantities. But I wouldn't 
> stress it at the moment . . .


We can already get at everything from the wedgie, by finding out what 
temperament it leads to and looking at those quantities.  If we can get 
"goodness" straight from the wedgie, that would save these calculations. 
  This is what geometric complexity might do.  And as it also gives us 
the straightness then, yes, it would mean we could skip straightness as 
an independent quantity.

The main thing is to get goodness of an incomplete wedgie.  Like we 
could find out that 2401:2400 and 3025:3024 work well together, and so 
keep looking for the next comma.  But maybe looking at the planar 
temperament would tell us that.  Which is like what I'm assuming about 
pairs of good equal temperaments giving good linear temperaments.


> Thanks for being you!


Well, sure, I do it all the time.


            Graham

Message: 8957 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 23:07:56

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Paul Erlich wrote:

>> Maybe someone can derive this 0.24999999999997 as a 1/4 
symbolically. 
>> I'd be very happy to see it.
> 

> 1.0014154337454717 you say?  Yes, I've derived it symbolically -- 
which 
> means I must have duplicated your method for tempering.  That magic 
> number is 2*log(81)/log(81*80).  The calculation goes something 
this:
> 
> The 2 generator is tempered as cents(2)[1+k]
> 
> The 3 generator is tempered as cents(3)[1-k]
> 
> The 5 generator is tempered as cents(5)[1+k]
> 
> where k is log(81/80)/log(81*80).  The + or - depends on whether 
factors 
> of this prime occur in the denominator or numerator respectively.


Yes, this is the method, as I recently explained here to Carl. But I 
didn't factor out [1+k] or [1-k] as multipliers -- that's a neat 
trick.


> So as 2 and 5 are just in quarter comma meantone, they must be 
stretched 
> by 1+k here.


Ah -- that's the secret. Good going!


> In quarter comma meantone, the 3 generator is tempered as
> 
> cents(3) - cents(81/80)/4
> 
> = cents(3) - cents(81)/4 + cents(80)/4
> = cents(3) - cents(3) + cents(80)/4
> = cents(80)/4
> 
> so the stretch from that to its new tempered value is
> 
> cents(3)[1-k]/[cents(80)/4]
> 
> = 4*cents(3)([1-k]/cents(80)
> = [1-k]cents(81)/cents(80)
> 
> Now we need to substitute in k, so that stretch becomes
> 
> [1-cents(81/80)/cents(81*80)]*cents(81)/cents(80)
> 
> (cents are a special case of log)
> 
> = [(cents(81*80) - cents(81/80))/cents(81*80)]*cents(81)/cents(80)
> = [2*cents(80)/cents(81*80)]*cents(81)/cents(80)
> = 2*cents(81)/cents(81*80)
> = 2*log(81)/log(81*80)
> 
> which is the same as 1+k, and you can work through that if you like.
> 
> 
>                   Graham


Nice work, Graham!

Message: 8958 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 23:09:07

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Paul Erlich wrote:
> 

>> Oh, you mean "heuristic", not "hypothesis" (the latter concerns 
PBs 
>> and DEs).
> 

> Yes, well, I typed the right thing into Google anyway :-)
> 

>> Hmm . . . I'm a bit busy right now, so can you work out an 
example, 
>> and show all the errors like I did with Top meantone here?
> 

> Meantone's the easy one.  81 is the odd part of 81/80.  So we need 
to 
> share the error of 81/80 amongst the factors of 81 according to 
their 
> respective weights.  As 3 is the only prime factor of 81, it takes 
all 
> the tempering, and each 3 gets a quarter of it, hence quarter comma 
> meantone.


Hmm . . . by *all* the errors, I meant for lots and lots of 
intervals, like I did.

Message: 8959 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 16:31:47

Subject: Re: Meantone reduced blocks

From: Carl Lumma

>> >ave you found any that don't reduce to something other than:
>> 
>> Meantone[12]
>> Diaschismic[12]
>> Augmented[12]
>> Diminished[12]
>> 
>> when tempered accordingly?
>
>Thirds.scl qualifies.


Is there an extra negation in your sentence there, Paul?

-Carl

Message: 8960 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 00:39:01

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

From: Paul Erlich

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

>>> Perhaps it has something to do with using this to get
>>> optimum generators for a linear temperament?
>> 

>> Well, that's exactly what this does (when the dimensionality is 
>> right), as I've illustrated already in a few cases.
>> 
>> Here's something new -- Top meantone is, it seems, exactly 1/4-
comma 
>> meantone (I get 0.24999999999997, but that's probably just 
rounding 
>> error) except a uniform (in cents, or log Hz) stretch of 
>> 1.00141543374547 is applied to all intervals . . .
> 

> Is the formula for that particularly hard?


Maybe someone can derive this 0.24999999999997 as a 1/4 symbolically. 
I'd be very happy to see it.


>>> And I don't understand your 'limitless' claim -- since p/q 
contains
>>> the factors it does and no others, one wouldn't expect its 
vanishing
>>> to effect
>> 
>> affect?
> 

> Yes, I think so...  :)
> 

>>> intervals different factors.
>> 

>> intervals with different factors? Well, 5:4 and 5:3 have *some* 
>> factors differing from those in the Pythagorean comma, yet both 
>> intervals are affected by its vanishing, in this scheme.
> 

> But not 7:5, right?


Right. Meanwhile, it seems that 81:80 vanishing leaves 6480:1 within 
a dust mite's excrement (which I'm allergic to, by the way) of 
vanishing . . .

Message: 8961 - Contents - Hide Contents

Date: Tue, 06 Jan 2004 00:44:35

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

From: Paul Erlich

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


>>>> claim -- since p/q 
> contains

>>>> the factors it does and no others, one wouldn't expect its 
> vanishing
>>>> to effect
>>> 
>>> affect?
>> 

>> Yes, I think so...  :)
>> 

>>>> intervals different factors.
>>> 

>>> intervals with different factors? Well, 5:4 and 5:3 have *some* 
>>> factors differing from those in the Pythagorean comma, yet both 
>>> intervals are affected by its vanishing, in this scheme.
>> 

>> But not 7:5, right?
> 

> Right. Meanwhile, it seems that 81:80 vanishing leaves 6480:1 
within 
> a dust mite's excrement (which I'm allergic to, by the way) of 
> vanishing . . .


Whoops, I meant "of being unaffected", not "of vanishing" . . .



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Message: 8962 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 14:21:20

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

From: Carl Lumma

>> > never understood this process,
>

>Solving a system of linear equations?

Uh-huh.


>> or what differentiates a period
>> from a generator.
>

>In our parlance, when we assume *octave-repetition*, the 'period' 
>will be the generator that generates the octave all by itself, while 
>the 'generator' (usually the smallest possible is chosen, such as 
>fourths for meantone) will produce all the other notes in the tuning 
>in conjunction with the period -- they form a basis.


Why are you assuming octave repetition, what does this assumption
amount to?

If 2 is in the map, one of the generators had better well generate
it.  If it isn't in the map, assuming octave repetition seems like
a bad idea to me.


>>>> And does the old method give different results when going from
>>>> 5-limit linear to 7-limit planar?
>>> 

>>> I believe so, though I can't remember the specifics.
>>> 

>>>> Or are you claiming the answer
>>>> is "no" when "old method" was minimax, and "yes" when it was
>>>> anything else?
>>> 

>>> If you mean Tenney-weighted minimax over all intervals, then this 
>>> could very well be, though I don't think that was actually one of 
>>> the "old" methods that were tried around here.
>> 

>> I'm still partial to rms over all the intervals,
>

>How can you do that? Does it even converge? Or do you not really 
>mean "all the intervals"?


I can't, and I mean all the odd-limit intervals including 2s, though
I suppose there may be difficulties in then allowing the size of the
2s to be a variable.


>> but somehow I
>> think those doing rms around here were not including the 2s.
>

>If you don't include all the intervals, but don't want to assume 
>octave-equivalence, you can use an integer limit, and I've posted 
>some integer-limit rms results on the tuning list and elsewhere.


Ok, that makes sense.  Odd-limit with tempered 2s means an infinite
number of intervals to optimize.  So I guess I've been asking for
integer limit all along.


>But 
>I don't like integer limit in comparison with Tenney limit, 
>especially a Tenney limit that you don't even have to specify (as 
>long as it's large enough to include all the primes you care about)!


What's a Tenney limit?

-Carl

Message: 8963 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 05:07:32

Subject: Re: Meantone reduced blocks

From: Gene Ward Smith

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

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

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

>>> Have you found any that don't reduce to something other than:
>>> 
>>> Meantone[12]
>>> Diaschismic[12]
>>> Augmented[12]
>>> Diminished[12]
>>> 
>>> when tempered accordingly?
>> 
>> Thirds.scl qualifies.
> 

> What were the unison vectors for that again?


Diesis and diesis--128/125 and 648/625.

Message: 8964 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 18:40:14

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

From: Carl Lumma

>>>> >hat do these equations look like?
>>> 
>>> For meantone,
>>> 

>>> prime2 = period;
>>> prime3 = period + generator;
>>> prime5 = 4*generator.
>>> 
>>> You can throw out any equation -- say the first.
>>> 
>>> so generator = .25*prime5,
>>> prime3 = period + .25*prime5,
>>> period = prime3 - .25*prime5.
>> 

>> Sure, I've done these hundreds of times.  But this is
>> just the map -- where are all the errors of all the
>> intervals?
>> 
>> -Carl
>

>Just add up the primes that make them up!


That sounds like TOP.  I'm talking about the old way.

-Carl

Message: 8965 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 22:21:32

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

From: Paul Erlich

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

>> Hiya Graham!  Let me rephrase the above.  Say I'm using unweighted
>> rms error over all the intervals in a given [odd] limit.  I want to
>> find the 5-limit linear temperament that minimizes this error, call
>> it Alex, and then I want to find the 7-limit planar temperament
>> that does the same, call it Ben.


Assuming they're same comma vanishes in both.


>> Now, are the 5-limit intervals in
>> Ben going to be different sizes than they are in Alex?


I believe so.


>> In TOP
>> temperament, the answer is no (I think).

Correct.

Message: 8966 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 06:39:36

Subject: The five or six Schisdia Scales

From: Gene Ward Smith

Five or six, because schisdia6 has already shown up under the name
syndia1, and before that as ramis and tamil_vi. It is the first 5-
limit Fokker block other than Pythagorean to show up under more than 
one rubric.

! schisdia1.scl
Schisdia 32805/32768 2048/2025 scale
12
!
16/15
9/8
6/5
81/64
4/3
64/45
3/2
8/5
27/16
3645/2048
256/135
2

! schisdia2.scl
Schisdia 32805/32768 2048/2025 scale
12
!
256/243
10/9
32/27
5/4
4/3
45/32
16384/10935
128/81
5/3
16/9
15/8
2

! schisdia3.scl
Schisdia 32805/32768 2048/2025 scale
12
!
135/128
4096/3645
32/27
5/4
4/3
45/32
3/2
128/81
2048/1215
16/9
15/8
2

! schisdia4.scl
Schisdia 32805/32768 2048/2025 scale
12
!
16/15
9/8
1215/1024
81/64
4/3
64/45
3/2
8/5
27/16
3645/2048
256/135
2

! schisdia5.scl
Schisdia 32805/32768 2048/2025 scale
12
!
135/128
9/8
1215/1024
512/405
4/3
64/45
3/2
405/256
27/16
16/9
256/135
2

! schisdia6.scl
Schisdia 32805/32768 2048/2025 scale ~ ramis tamil_vi syndia1
12
!
16/15
9/8
6/5
81/64
4/3
64/45
3/2
8/5
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2

Message: 8967 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 22:26:58

Subject: Hermite normal form

From: Gene Ward Smith

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


> As usual, however, nowhere on this list can I find any explanation
> of hermite reduction.  Not what it is, not why we'd care, and not
> how to calculate it.


Both Maple and Mathematica implement Hermite reduction, so one thing 
you can do is to read the documentation for Maple's "ihermite" and 
Mathematica's "HermiteNormalForm". Here's World of Math on it, 
unfortunately restricted to square matricies, which is not what we 
would be using:

Hermite Normal Form -- from MathWorld * [with cont.] 

I don't know if Mathematica only reduces square matricies, but this 
entry suggests that it might. Maple's "ihermite" works the way we 
want it to, and works for both listlists and arrays, which is 
convenient.

Message: 8968 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 19:17:04

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

From: Carl Lumma

>> >hat sounds like TOP.  I'm talking about the old way.
>

>Oh. The old way, you start with the mapping, and then solve for the 
>period and generator, and then minimize your error function (which is 
>over some finite set of intervals) by varying the period & generator, 
>or in octave-equivalent cases, just the generator. You can use 
>calculus


Aha, I knew it!  Calculus!  :)


>and express the error function in terms of the generator 
>size, take the derivative,

ok...

>set that equal to zero, and solve


Lost me here.  The derivative itself is a curve, unless the
error/generator function is a straight line or something.

Wait -- are you saying that once the error fuctio starts
going up it'll never go down again?

Oh, and if we're doing integer limit don't we need two
generators?


>-- works great for sum-squared error (p=2), weighted or unweighted.


Good, that's all I want.  I've got enough software to put my eye
out with, I ought to be able to set this up.  By the way, this now
includes Matlab, if you'd prefer to illustrate with code.

-Carl

Message: 8969 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 19:43:35

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

From: Paul Erlich

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

>>>> Now, for all primes r,
>>>> 
>>>> If p contains any factors of r, the r-rungs in the lattice 
(which 
>>>> have length log2(r)) are shrunk from
>>>> cents(r)
>>>> to
>>>> cents(r) - log2(r)*cents(p/q)/log2(p*q).
>>>> If q contains any factors of 2, they are instead stretched to
>>>> cents(r) + log2(r)*cents(p/q)/log2(p*q).
>>> 

>>> Thanks.  I understand this 100%.  But I don't understand what's
>>> new.
>> 

>> Where have you seen this before?
> 

> I guess in my head.
> 

>>> Perhaps it has something to do with using this to get
>>> optimum generators for a linear temperament?
>> 

>> Well, that's exactly what this does (when the dimensionality is 
>> right), as I've illustrated already in a few cases.
> 

> What if the generator isn't a just interval?  Then isn't it still
> the same kind of multivariable optimization that you guys have been
> using all along?


I didn't make any assumptions about what the generator was above. The 
same formula works for any generator, and even when there is no 
generator, as is the case for 7-limit and above.


>>> And I don't understand your 'limitless' claim -- since p/q 
contains
>>> the factors it does and no others, one wouldn't expect its 
vanishing
>>> to effect
> 

> What I mean is, when extending meantone to a 7-limit mapping, it
> will naturally implicate different commas and change the optimal
> generator a bit, same as before.


Yes, and there are different choices as to which commas to use to 
extend meantone to a 7-limit linear temperament. But I wasn't talking 
about that. I was talking about tempering out a single comma, which 
would lead to a planar temperament in the 7-limit, etc.


> But to claim it doesn't require a limit
> when it's currently limited to linear temperaments in the 5-limit...


No, it's simply limited to temperaments of codimension 1. Though I've 
only charted the 5-limit commas, the exact same method works for any 
commas, and I'll be producing a 7-limit comma chart for Herman soon.

Message: 8970 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 22:28:33

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

From: Paul Erlich

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

>>> I never understood this process,
>> 

>> Solving a system of linear equations?
> 
> Uh-huh.


Well, the easiest way to understand is to solve one equation for one 
variable, plug that solution into the other variables so that you've 
eliminated one variable entirely, and repeat until you're done.


>>> or what differentiates a period
>>> from a generator.
>> 

>> In our parlance, when we assume *octave-repetition*, the 'period' 
>> will be the generator that generates the octave all by itself, 
while 
>> the 'generator' (usually the smallest possible is chosen, such as 
>> fourths for meantone) will produce all the other notes in the 
tuning 
>> in conjunction with the period -- they form a basis.
> 

> Why are you assuming octave repetition, what does this assumption
> amount to?


That you'll have the same pitches in each (possibly tempered) octave.


> If 2 is in the map, one of the generators had better well generate
> it.  If it isn't in the map, assuming octave repetition seems like
> a bad idea to me.


Any recent cases where you'd prefer not to see 2 in the map?


>>>>> And does the old method give different results when going from
>>>>> 5-limit linear to 7-limit planar?
>>>> 

>>>> I believe so, though I can't remember the specifics.
>>>> 

>>>>> Or are you claiming the answer
>>>>> is "no" when "old method" was minimax, and "yes" when it was
>>>>> anything else?
>>>> 

>>>> If you mean Tenney-weighted minimax over all intervals, then 
this 
>>>> could very well be, though I don't think that was actually one 
of 
>>>> the "old" methods that were tried around here.
>>> 

>>> I'm still partial to rms over all the intervals,
>> 

>> How can you do that? Does it even converge? Or do you not really 
>> mean "all the intervals"?
> 

> I can't, and I mean all the odd-limit intervals including 2s,


There are an infinite number of those, but if the octave is fixed at 
1200 cents, you only need one member of each class, and then you have 
a finite list of intervals, so you do get convergence.


> though
> I suppose there may be difficulties in then allowing the size of the
> 2s to be a variable.

Correct.

> What's a Tenney limit?


If the limit is L, it's all reduced ratios n/d such that n*d<=L (or 
log(n*d)<=L, whatever).

Message: 8971 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 19:45:01

Subject: Re: Meantone reduced blocks

From: Paul Erlich

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

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

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

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

>>>> Have you found any that don't reduce to something other than:
>>>> 
>>>> Meantone[12]
>>>> Diaschismic[12]
>>>> Augmented[12]
>>>> Diminished[12]
>>>> 
>>>> when tempered accordingly?
>>> 
>>> Thirds.scl qualifies.
>> 

>> What were the unison vectors for that again?
> 

> Diesis and diesis--128/125 and 648/625.


But doesn't that reduce to Augmented[12] when 128/125 is tempered out 
and Diminished[12] when 648/625 is tempered out??

Message: 8972 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 22:31:29

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

From: Graham Breed

Carl Lumma wrote:


> Hiya Graham!  Let me rephrase the above.  Say I'm using unweighted
> rms error over all the intervals in a given prime limit.  I want to
> find the 5-limit linear temperament that minimizes this error, call
> it Alex, and then I want to find the 7-limit planar temperament
> that does the same, call it Ben.  Now, are the 5-limit intervals in
> Ben going to be different sizes than they are in Alex?  In TOP
> temperament, the answer is no (I think).

Hello!


How would an unweighted, unbounded RMS error work?  The advantage of the 
weighting is that more complex intervals get lower weights, and so the 
weighted error stays roughly constant.  Hence you can impose a weighted 
minimax over all intervals within a given prime limit.  The interesting, 
and slightly unexpected, thing about TOP is that it goes straight to 
this weighted minimax.

Oh, you meant odd limit.  Well, I'll leave that paragraph in anyway.

Still to the question.  Yes, I think the answer is no.  If you define a 
7-limit planar temperament by a 5-limit comma, the TOP method will give 
you a 5-limit linear temperament along with a just 7:4.  And that should 
also be the same temperament you'd get by finding the weighted minimax 
for that planar temperament by any other method.


                Graham

Message: 8973 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 20:03:53

Subject: Re: Meantone reduced blocks

From: Gene Ward Smith

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


> But doesn't that reduce to Augmented[12] when 128/125 is tempered 
out 
> and Diminished[12] when 648/625 is tempered out??


Ooops. Had my wiring crossed.

Message: 8974 - Contents - Hide Contents

Date: Wed, 07 Jan 2004 20:07:27

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

From: Carl Lumma

>>> >nd express the error function in terms of the generator 
>>> size, take the derivative,
>> 
>> ok...
>> 

>>> set that equal to zero, and solve
>> 

>> Lost me here.  The derivative itself is a curve,
>

>Right, and where it meets the x-axis is where it equals zero.


That's where the original function is flat, but how do we
know the original function isn't flat at multiple places?


>> Oh, and if we're doing integer limit don't we need two
>> generators?
>

>We need two generators if we're talking about a 2D temperament -- 
>either a planar temperament with octave-equivalence assumed, or a 
>linear temperament where we can vary the octave (or period) as well 
>as the generator.


I'm talking about linear temperaments now, strictly.  And by
"integer limit", I mean variable octave, and I've been calling
the period a generator for, oh, over a year.


>>> -- works great for sum-squared error (p=2), weighted or unweighted.
>> 

>> Good, that's all I want.  I've got enough software to put my eye
>> out with, I ought to be able to set this up.  By the way, this now
>> includes Matlab, if you'd prefer to illustrate with code.
>

>Wow. Do you have the optimization toolbox?


It looks like it.  I just ran a "Large-scale unconstrained nonlinear
minimization" demo.

-Carl

-Carl

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