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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.

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.]

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

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?

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?

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

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

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.

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

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

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.

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).

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

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

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

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

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

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 . . .

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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