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Message: 7128

Date: Tue, 29 Jul 2003 22:42:18

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
> 
> > What wasn't clear?
> 
> You haven't given a general formula, only a lot of specific 
formulae 
> involving real numbers you don't derive and indices that only work 
for 
> your particular implementation.

Not true any more, and I think I posted this. Here is the Maple code 
for 7-limit geometric complexity:

gc7 := proc(l)
# l is 7-limit wedgie
sqrt(evalf(1/4*(-ln(5)^4+4*ln(5)^2*ln(7)^2)*l[1]^2
+1/4*(-ln(3)^4+4*ln(3)^2*ln(7)^2)*l[2]^2
+1/4*(-ln(3)^4+4*ln(3)^2*ln(5)^2)*l[3]^2
+1/2*(ln(3)^2*ln(5)^2-2*ln(3)^2*ln(7)^2)*l[1]*l[2]
-1/2*ln(3)^2*ln(5)^2*l[1]*l[3]+
1/2*(ln(3)^4-2*ln(3)^2*ln(5)^2)*l[2]*l[3])) end:

Here is 11-limit linear:

sqrt(evalf((-1/4*ln(5)^2*ln(7)^4-1/4*ln(11)^2*ln(5)^4+ln(7)^2*ln(11)
^2*ln(5)^2)*l[1]^2
+(ln(11)^2*ln(7)^2*ln(3)^2-1/4*ln(3)^4*ln(11)^2-1/4*ln(3)^2*ln(7)^4)*l
[2]^2
+(ln(3)^2*ln(5)^2*ln(11)^2-1/4*ln(3)^2*ln(5)^4-1/4*ln(3)^4*ln(11)^2)*l
[3]^2
+(ln(7)^2*ln(3)^2*ln(5)^2-1/4*ln(3)^2*ln(5)^4-1/4*ln(7)^2*ln(3)^4)*l
[4]^2
+(1/2*ln(3)^2*ln(5)^2*ln(11)^2-ln(11)^2*ln(7)^2*ln(3)^2+1/4*ln(3)^2*ln
(7)^4)*l[1]*l[2]
+(-1/2*ln(3)^2*ln(5)^2*ln(11)^2+1/4*ln(7)^2*ln(3)^2*ln(5)^2)*l[1]*l[3]
-1/4*ln(7)^2*ln(3)^2*ln(5)^2*l[1]*l[4]
+(-ln(3)^2*ln(5)^2*ln(11)^2+1/2*ln(7)^2*ln(3)^2*ln(5)^2-1/4*ln(7)^2*ln
(3)^4+
1/2*ln(3)^4*ln(11)^2)*l[2]*l[3]+(-1/2*ln(7)^2*ln(3)^2*ln(5)^2+1/4*ln
(7)^2*ln(3)^4)*l[2]*l[4]
+(-ln(7)^2*ln(3)^2*ln(5)^2+1/2*ln(3)^2*ln(5)^4+1/4*ln(7)^2*ln(3)^4)*l
[3]*l[4])) end:

11-limit planar:

gpc11 := proc(l)
# l is 11-limit planar wedgie
sqrt(evalf((-1/4*ln(7)^4+ln(7)^2*ln(11)^2)*l[1]^2+(-1/4*ln(5)^4+ln(5)
^2*ln(11)^2)*l[2]^2
+(-1/4*ln(5)^4+ln(5)^2*ln(7)^2)*l[3]^2+(-1/4*ln(3)^4+ln(3)^2*ln(11)^2)
*l[4]^2
+(-1/4*ln(3)^4+ln(3)^2*ln(7)^2)*l[5]^2+(-1/4*ln(3)^4+ln(3)^2*ln(5)^2)
*l[6]^2
-(-1/2*ln(5)^2*ln(7)^2+ln(5)^2*ln(11)^2)*l[1]*l[2]-1/2*ln(5)^2*ln(7)
^2*l[1]*l[3]
+(-1/2*ln(3)^2*ln(7)^2+ln(3)^2*ln(11)^2)*l[1]*l[4]+1/2*ln(3)^2*ln(7)
^2*l[1]*l[5]
+(1/2*ln(3)^2*ln(5)^2-1/2*ln(3)^2*ln(7)^2)*l[1]*l[6]-(-1/2*ln(5)^4+ln
(5)^2*ln(7)^2)*l[2]*l[3]
-(-1/2*ln(3)^2*ln(5)^2+ln(3)^2*ln(11)^2)*l[2]*l[4]+(-ln(3)^2*ln(5)
^2+ln(3)^2*ln(7)^2)*l[2]*l[5]
+1/2*ln(3)^2*ln(5)^2*l[2]*l[6]-(-1/2*ln(3)^2*ln(5)^2+ln(3)^2*ln(7)^2)
*l[3]*l[5]
-1/2*ln(3)^2*ln(5)^2*l[3]*l[6]-(-1/2*ln(3)^4+ln(3)^2*ln(7)^2)*l[4]*l
[5]
+(-1/2*ln(3)^4+ln(3)^2*ln(5)^2)*l[4]*l[6]-(-1/2*ln(3)^4+ln(3)^2*ln(5)
^2)*l[5]*l[6])) end:


It's unfortunately true that these results are not immediate from the 
definition of geometric complexity, but must themselves be computed, 
so only for 5-limit is this straightforward.

> If I had worked out your numbering rule, I've forgotten it now.
> 
> Every time you try to explain something, you bring in more jargon 
terms 
> that I don't understand (I can't speak for anybody else).
> 
> The word "metric" in particular is something that's important but 
you 
> haven't defined.

I've posted this before; it's standard math:

Metric -- from MathWorld * [with cont.] 

> You keep missing out important steps in explanations, like the need 
to 
> take the complement at certain points when using wedge products.

As I've explained before, the way I wrote my Maple code I don't need 
to. All that is built into the functions themselves.


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Message: 7129

Date: Tue, 29 Jul 2003 06:39:46

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

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

Maybe Gene can prod us to figure out geometric complexity.

What wasn't clear?

> Yahoo groups: /tuning-math/message/5546 * [with cont.] 
> Yahoo groups: /tuning-math/message/5598 * [with cont.] 
> Yahoo groups: /tuning-math/message/5671 * [with cont.] 
> Yahoo groups: /tuning-math/message/5692 * [with cont.] 


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Message: 7130

Date: Tue, 29 Jul 2003 10:09:10

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Gene Ward Smith wrote:

> What wasn't clear?

You haven't given a general formula, only a lot of specific formulae 
involving real numbers you don't derive and indices that only work for 
your particular implementation.

If I had worked out your numbering rule, I've forgotten it now.

Every time you try to explain something, you bring in more jargon terms 
that I don't understand (I can't speak for anybody else).

The word "metric" in particular is something that's important but you 
haven't defined.

You keep missing out important steps in explanations, like the need to 
take the complement at certain points when using wedge products.


                               Graham


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Message: 7133

Date: Wed, 30 Jul 2003 08:08:55

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >I would like to know which temperment/comma connects 12, 47, 35, 
and 
> >> >23 ets on zoomr.gif. How could I calculate these for myself? 
Thanks!
> >> >If this is a new one, could I name it? What is the 5-limit 
vector?
> >> 
> >> The comma is 6561/6250, according to Gene's maple, if I did it 
right.
> >
> >I get the same. You get a temperament with a generator the size of 
a
> >semitone, five of which give a fourth, and eight of which give a 
minor
> >sixth, which obviously is compatible with 12-et.
> 
> So what I'm missing is... the heuristic says temperaments based on
> this comma will be bad.

which heuristic -- the one for complexity or the one for error?

> And indeed, 23, 47 don't seem to be very
> good.  But 12 is good.  So what gives?

the heuristic for error applies to the best choice, in some sense. so 
something close to 12. but i don't know how you came up with "bad" ~= 
"good" inequality here. could you try to be more quantitative?


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Message: 7135

Date: Wed, 30 Jul 2003 08:20:44

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

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

> >For a linear temperament, the number of consonances is (roughly)
> >the number of notes minus the complexity.  So n-i is a constant.
> >If you then plot n/i, you can replace n with i+c to get
> >(i+c)/i = 1+c/i.  As i tends to infinity, this tends to 1, which
> >is as small as it gets.  So your measure would depend on how
> >large n is allowed to get.
> 
> Ok, I follow your reasoning but I'm not sure what your conclusion
> is.  I think we'd limit n to the number of notes in the Fokker
> block.

what's "the" fokker block? you mean "a" fokker block?


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Message: 7136

Date: Wed, 30 Jul 2003 11:28:31

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>Thanks for all the info. I have a ton of libraries, but no temper.py
>I downloaded Python a while ago.

I was confused by Graham calling it a library too.  It's *his*
contribution, available from his web site.

BTW, Python 2.3 final was released 2 days ago.  But don't try to
put a whitespace in the install path on Windows!

-Carl


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Message: 7137

Date: Wed, 30 Jul 2003 01:21:57

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>>>>The comma is 6561/6250,
//
>>>You get a temperament with a generator the size of a semitone,
>>>five of which give a fourth, and eight of which give a minor
>>>sixth, which obviously is compatible with 12-et.
>> 
>>So what I'm missing is... the heuristic says temperaments
>>based on this comma will be bad.
>
>which heuristic -- the one for complexity or the one for error?

Error.  n-d seems large considering d.  I admit I just did this
by eye, without actually calculating the heuristic.

>> And indeed, 23, 47 don't seem to be very
>> good.  But 12 is good.  So what gives?
>
>the heuristic for error applies to the best choice, in some sense.

Ah.

-Carl


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Message: 7138

Date: Wed, 30 Jul 2003 23:13:34

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >not only that, but you're missing a unison vector! a linear 
> >temperament has one fewer defining unison vectors than a 
periodicity 
> >block in the same space.
> 
> Only one?

yes.

> Doesn't the number missing depend on the space?

no.


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Message: 7139

Date: Wed, 30 Jul 2003 01:24:05

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> >For a linear temperament, the number of consonances is (roughly)
>> >the number of notes minus the complexity.  So n-i is a constant.
>> >If you then plot n/i, you can replace n with i+c to get
>> >(i+c)/i = 1+c/i.  As i tends to infinity, this tends to 1, which
>> >is as small as it gets.  So your measure would depend on how
>> >large n is allowed to get.
>> 
>> Ok, I follow your reasoning but I'm not sure what your conclusion
>> is.  I think we'd limit n to the number of notes in the Fokker
>> block.
>
>what's "the" fokker block? you mean "a" fokker block?

A, but clearly straightness or something similar needs to be invoked.

-Carl


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Message: 7140

Date: Wed, 30 Jul 2003 16:17:22

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> a linear temperament has one fewer defining unison vectors than a 
>> periodicity block in the same space.

A linear temperament has the same number of uvs as a block, but
with all but one of them tempered out.  Now, how are you defining
"defining"?

-Carl


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Message: 7141

Date: Wed, 30 Jul 2003 08:36:22

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >For a linear temperament, the number of consonances is (roughly)
> >> >the number of notes minus the complexity.  So n-i is a constant.
> >> >If you then plot n/i, you can replace n with i+c to get
> >> >(i+c)/i = 1+c/i.  As i tends to infinity, this tends to 1, which
> >> >is as small as it gets.  So your measure would depend on how
> >> >large n is allowed to get.
> >> 
> >> Ok, I follow your reasoning but I'm not sure what your conclusion
> >> is.  I think we'd limit n to the number of notes in the Fokker
> >> block.
> >
> >what's "the" fokker block? you mean "a" fokker block?
> 
> A, but clearly straightness or something similar needs to be 
invoked.
> 
> -Carl

not only that, but you're missing a unison vector! a linear 
temperament has one fewer defining unison vectors than a periodicity 
block in the same space.


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Message: 7142

Date: Wed, 30 Jul 2003 23:34:15

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> a linear temperament has one fewer defining unison vectors than 
a 
> >> periodicity block in the same space.
> 
> A linear temperament has the same number of uvs as a block, but
> with all but one of them tempered out.

no, that's not a linear temperament, that's an MOS scale (or what we 
used to think was called an MOS scale).


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Message: 7143

Date: Wed, 30 Jul 2003 01:41:31

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>not only that, but you're missing a unison vector! a linear 
>temperament has one fewer defining unison vectors than a periodicity 
>block in the same space.

Only one?  Doesn't the number missing depend on the space?

-Carl


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Message: 7144

Date: Wed, 30 Jul 2003 16:43:17

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> A linear temperament has the same number of uvs as a block,
>> but with all but one of them tempered out.
>
>no, that's not a linear temperament, that's an MOS scale

Whoa.  Is there a friendly guide to temperaments somewhere?

-Carl


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Message: 7145

Date: Wed, 30 Jul 2003 09:44:43

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Carl Lumma wrote:

> Only one?  Doesn't the number missing depend on the space?

A linear temperament always needs one unison vector less than a 
periodicity block.  An equal temperament needs the same number, hence 
Fokker's 31 note periodicity blocks.  A planar temperament needs two 
less, and so on.

I say "needs" because you can always add unison vectors if you start 
with more than one.  The minimal set is linearly independent.


                           Graham


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Message: 7146

Date: Wed, 30 Jul 2003 16:45:14

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> Only one?  Doesn't the number missing depend on the space?
>
>A linear temperament always needs one unison vector less than a 
>periodicity block.  An equal temperament needs the same number, hence 
>Fokker's 31 note periodicity blocks.  A planar temperament needs two 
>less, and so on.

So this formulation does depend on the space, since the number
needed for a block depends on the space!

>I say "needs" because you can always add unison vectors if you start 
>with more than one.  The minimal set is linearly independent.

Yeah, I think I got that.

-Carl


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Message: 7147

Date: Wed, 30 Jul 2003 12:41:56

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

paulhjelmstad wrote:

> Thanks. Where can I get the Library for Python? I have the command 
> shell but get an error running the above command.

"The library" is the file temper.py, linked as "script" from the source 
code page.  If it's in your Python path, you import it with the command 
"import temper".  If that gives an error, you either need to read the 
documentation on the Python path, or follow this recipe (which depends 
on where temper.py is):

 >>> import sys
 >>> sys.path.append(r'd:/net/microtonal')
 >>> import temper

The first "test script" on the source code page gives you examples of 
how to use the library.  The dir and help (depending on the Python 
version) builtin functions can be useful:

 >>> dir(temper)
['BestET', 'EqualTemperament', 'LinearTemperament', 'OddLimit', 
'PrimeDiamond', 'PrimeET', 'Temperament', 'TemperamentException', 
'TemperamentList', 'TonalityDiamond', 'UserDict', 'UserList', 
'WedgableRatio', 'Wedgie', 'WedgieET', '__builtins__', '__doc__', 
'__file__', '__name__', 'alternativeMappings', 'combinations', 
'defaultFigureOfDemerit', 'dotprod', 'euclidianGCD', 'exp', 'factorize', 
'factorizeRatio', 'getCombinations', 'getDimensions', 'getDivisors', 
'getGenerator', 'getLimitedETs', 'getLinearTemperaments', 'getRatio', 
'hcf', 'intervalCompare', 'limit11', 'limit13', 'limit15', 'limit17', 
'limit19', 'limit21', 'limit5', 'limit7', 'limit9', 
'linearTemperamentsFromIntervals', 
'linearTemperamentsFromUnisonVectors', 'log', 'log2', 'makeWedgie', 
'new', 'nint', 'normalizeInterval', 'operator', 'primeNumbers', 
'primes', 'ratioMatch', 're', 'readVectors', 'reduceWithinTritone', 
'simplestOddLimit', 'sqrt', 'string', 'temperOut', 
'uniqueWithinTritone', 'unisonVectorsFromIntervals', 'wedgeEquivalent', 
'wedgeProduct']
 >>> help(temper.BestET)
Help on function BestET in module temper:

BestET(nNotes, consonances, cutoff=1.0)


---

To understand Python itself, see the tutorial at

Python Tutorial * [with cont.]  (Wayb.)

and you probably downloaded a copy of that with the interpreter.


                     Graham


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Message: 7148

Date: Wed, 30 Jul 2003 23:45:10

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> A linear temperament has the same number of uvs as a block,
> >> but with all but one of them tempered out.
> >
> >no, that's not a linear temperament, that's an MOS scale
> 
> Whoa.

you should have known that, that's the Hypothesis.

> Is there a friendly guide to temperaments somewhere?
> 
> -Carl

you've probably seen anything i could suggest already . . .


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Message: 7149

Date: Wed, 30 Jul 2003 23:45:58

Subject: Re: Creating a Temperment /Comma

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Only one?  Doesn't the number missing depend on the space?
> >
> >A linear temperament always needs one unison vector less than a 
> >periodicity block.  An equal temperament needs the same number, 
hence 
> >Fokker's 31 note periodicity blocks.  A planar temperament needs 
two 
> >less, and so on.
> 
> So this formulation does depend on the space, since the number
> needed for a block depends on the space!

but the *difference* is always 1, regardless of the space. it's 
called the "codimension".


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