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Message: 7100

Date: Fri, 25 Jul 2003 23:05:22

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad"
<paul.hjelmstad@u...> wrote:

> > Thanks. (3)^8/(5)^5*(2) Hmm. 84 cents? I christen it "sloppyjoe"
> Rats. It's been taken. Can't name it. It's "Diaschizoid"

Taken by what?


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Message: 7101

Date: Fri, 25 Jul 2003 16:14:00

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

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

-Carl


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Message: 7103

Date: Fri, 25 Jul 2003 16:23:20

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>Well, I can't find it so I doubt it's much good. Thanks. If I bought 
>Maple and installed it, could I possibly run these calculations 
>myself?

I'm sure Gene will give you the code.  Maybe he'll even post it
on his web site.

By the Way, Paaauuul (Erlich), I'm told Matlab has the Maple engine
built into it!!

-Carl


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Message: 7105

Date: Fri, 25 Jul 2003 23:32:18

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

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

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

As a linear temperament, it isn't very interesting, because its
complexity is too high. As a means of tempering 12 notes, it becomes
much more interesting. Looking at temperaments dominated (as this one
is) by 12 strikes me as worthwhile from that point of view. I think
I'll check out this idea of semitones of around 101.5 cents for 12 notes.


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Message: 7106

Date: Fri, 25 Jul 2003 23:33:52

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Well, I can't find it so I doubt it's much good. Thanks. If I bought 
> >Maple and installed it, could I possibly run these calculations 
> >myself?
> 
> I'm sure Gene will give you the code.  Maybe he'll even post it
> on his web site.

I'm planning too. Hope no one laughs.


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Message: 7107

Date: Fri, 25 Jul 2003 23:41:53

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "paulhjelmstad"
<paul.hjelmstad@u...> wrote:

(But I can't find 
> anything for 12, 25, 37, 49 ets etc)

This has comma 262144/253125. Once again the generator is about a
semitone and five of them give a fourth. This time, however, the
semitone is a little smaller than 12-et (about 98.3 cents) and four of
them give you a major third. Evidently these two belong together and
should get cute matching names.


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Message: 7108

Date: Fri, 25 Jul 2003 16:44:23

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>As a linear temperament, it isn't very interesting, because its
>complexity is too high. As a means of tempering 12 notes, it becomes
>much more interesting.

Is this because there's a T[n] comma associated with stopping at
12?  Or...?

By the way, what wound up being your favorite complexity measure
and why?  The last 5-limit list I have from you uses "geometric
complexity"... which is...  Oh, that's the really hard-to-understand
Euclidean distance-on-the-lattice-to-the-comma thing, isn't it?
 
Meanwhile, Dave and Graham, are you guys still using map-based
complexity?  Doesn't this run into problems because there are
many different ways to write the map for a given temperament?

Me, I like the idea of number of notes.  But then we have to
consider the effects of the comma formed between the ends of
the chain.  Has anyone ever looked at error as a function of
chain length for each temperament?

-Carl


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Message: 7109

Date: Fri, 25 Jul 2003 19:34:19

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>>>As a linear temperament, it isn't very interesting, because its
>>>complexity is too high. As a means of tempering 12 notes, it
>>>becomes much more interesting.
>> 
>> Is this because there's a T[n] comma associated with stopping at
>> 12?  Or...?
//
>Just as with meantone, we can temper 12 notes, or we can allow
>ourselves to take something written in 12 notes and extend it to
>more using the temperament;

How does one "take something written in 12 notes and extend it with
the temperament"?

>>By the way, what wound up being your favorite complexity
>>measure and why? 
>
>I like geometric complexity because it works for all regular
>temperaments.

Such as planar and higher temperaments.  But the 'how many notes'
complexity obviously generalizes to these as well (in the linear
case it's equivalent to Dave/Graham complexity).

>> Me, I like the idea of number of notes.  But then we have to
>> consider the effects of the comma formed between the ends of
>> the chain.  Has anyone ever looked at error as a function of
>> chain length for each temperament?
>
>Aren't you conflating temperaments and scales?

I understand that temperaments are abstract Things, but they
ultimately must be expressed as scales to be used.  If geometric
complexity is so good, the T[n] stuff wouldn't have been such a
surprise (unless, I suppose, consonances formed by way of the
extra "wolf" comma necessarily break consistency -- do they?).

In my view, musically complexity has to boil down to 'how many
notes get me how many consonances'.  Say n/i where n is notes
and i is the number of consonant dyads.  Dave/Graham get rid of
the denominator by standardizing i to 'a complete n-ad' (Dave
once showed that for linear temperaments, otonal and utonal n-ads
always come in pairs -- I wonder if this is true for planar
temperaments...).  I suggest we plot n/i for temperament T for
all n up to some large number, and report the minima.  If this
is a bad idea I'd like to know why.

-Carl


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Message: 7110

Date: Sat, 26 Jul 2003 13:14:41

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> Such as planar and higher temperaments.  But the 'how many notes'
>> complexity obviously generalizes to these as well (in the linear
>> case it's equivalent to Dave/Graham complexity).
>
>What about equal temperaments?

For an n-et, one can always think of the generators as 1200 and
1200/n.  Then it works the same as for linear temperaments.  Or
isn't that what you do?

>> In my view, musically complexity has to boil down to 'how many
>> notes get me how many consonances'.  Say n/i where n is notes
>> and i is the number of consonant dyads.  Dave/Graham get rid of
>> the denominator by standardizing i to 'a complete n-ad' (Dave
>> once showed that for linear temperaments, otonal and utonal n-ads
>> always come in pairs -- I wonder if this is true for planar
>> temperaments...).  I suggest we plot n/i for temperament T for
>> all n up to some large number, and report the minima.  If this
>> is a bad idea I'd like to know why.
>
>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.  At that n, the complexity would be expected to match the
heuristic (for linear temperaments) and geometric complexity (for
everything).  But when n gets smaller we need a way to quantify
what happens.  It was probably wrong of me to suggest this be some
sort of temperament complexity.  And I'm wondering if intervals
achieved through use of the "wolf" break consistency, or whatever
Gene calls it for regular temperaments.

>For an equal temperament, n can only ever take one value, and
>n/i=1.

How do you figure the number of consonant intervals equals the
number of notes in the et?

>For a planar temperament, the number of consonance depends on what 
>generators you choose, and how you construct the scale from them.
>You can get more otonal than utonal if you like.

Aha.

>Taking 5-limit JI as the canonical planar scale, C-E-G gives 1
>more otonality than utonality.  And C-D-E-G-G#-B has 2 more.

Do such blocks necessarily involve unison vectors that are larger
than their smallest 2nd?

>But you should get the same value for complexity whether you use
>otonal or utonal.

Excactly -- which is why they apparently both must be counted
for planar temperaments.

-Carl


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Message: 7111

Date: Sat, 26 Jul 2003 01:19:58

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

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

> >As a linear temperament, it isn't very interesting, because its
> >complexity is too high. As a means of tempering 12 notes, it becomes
> >much more interesting.
> 
> Is this because there's a T[n] comma associated with stopping at
> 12?

Only if you think a comma would be interesting.

  Or...?

There's a historical importance to 12 which is, of course, obvious.
There are just two ways to generate a cyclic group of order 12--one
gives the fourth/fifth generators we usually focus on, the other is
the semitone. Just as with meantone, we can temper 12 notes, or we can
allow ourselves to take something written in 12 notes and extend it to
more using the temperament; it's an idea worth exploring, at any rate.

> By the way, what wound up being your favorite complexity measure
> and why? 

I like geometric complexity because it works for all regular temperaments.

 The last 5-limit list I have from you uses "geometric
> complexity"... which is...  Oh, that's the really hard-to-understand
> Euclidean distance-on-the-lattice-to-the-comma thing, isn't it?

Fraid so.

> Me, I like the idea of number of notes.  But then we have to
> consider the effects of the comma formed between the ends of
> the chain.  Has anyone ever looked at error as a function of
> chain length for each temperament?

Aren't you conflating temperaments and scales?


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Message: 7112

Date: Sat, 26 Jul 2003 06:01:46

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

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

> How does one "take something written in 12 notes and extend it with
> the temperament"?

These are 5-limit temperaments, so you should concentrate on getting 
good triads.


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Message: 7113

Date: Sat, 26 Jul 2003 12:26:36

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Carl Lumma wrote:

> Such as planar and higher temperaments.  But the 'how many notes'
> complexity obviously generalizes to these as well (in the linear
> case it's equivalent to Dave/Graham complexity).

What about equal temperaments?

> In my view, musically complexity has to boil down to 'how many
> notes get me how many consonances'.  Say n/i where n is notes
> and i is the number of consonant dyads.  Dave/Graham get rid of
> the denominator by standardizing i to 'a complete n-ad' (Dave
> once showed that for linear temperaments, otonal and utonal n-ads
> always come in pairs -- I wonder if this is true for planar
> temperaments...).  I suggest we plot n/i for temperament T for
> all n up to some large number, and report the minima.  If this
> is a bad idea I'd like to know why.

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.

For an equal temperament, n can only ever take one value, and n/i=1.

For a planar temperament, the number of consonance depends on what 
generators you choose, and how you construct the scale from them.  You 
can get more otonal than utonal if you like.  Taking 5-limit JI as the 
canonical planar scale, C-E-G gives 1 more otonality than utonality. 
And C-D-E-G-G#-B has 2 more.  But you should get the same value for 
complexity whether you use otonal or utonal.  If you standardize on a 
parallelogram, like C-E-G-B, than you get equal numbers of each chord.

That's assuming you take the fifth and third as generators.  With a tone 
and semitone, it'd take 5 notes to get 1 triad, and 6 notes to get 1 of 
each triad (which gets you 2 of 1 kind).  I haven't looked into planar 
temperaments yet, so I don't know if there's an obvious way to get the 
optimum generators.

I think you still get n/i tends to 1 as n tends to infinity.


                           Graham


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Message: 7114

Date: Sat, 26 Jul 2003 12:52:03

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

paulhjelmstad 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?

Go to

Temperament Finder * [with cont.]  (Wayb.)

and enter 12 and 47 for the ets, and 5 for the limit.  That gives you 
the mapping

[(1, 0), (2, -5), (3, -8)]

It doesn't show the comma because I don't see the point, and they aren't 
that easy to calculate for some temperaments.  But for the 5-limit case 
you can work it out from the mapping.  It must involve 3**8 and 5**5 
with opposite signs, from the octave equivalent mapping (-5, -8).  You 
then multiply (x, 8, -5) by the period part of the mapping to get x + 
2*8 - 3*5.  For a unison vector, this should be zero, so x = -2*8 + 3*5 
= 15-16 = -1.  So the comma is (-1 8 -5) or 2**(-1) * 3**8 * 5**(-5) = 
6561:6250.

There are different tools runnable online at 
Linear Temperament Finding Home * [with cont.]  (Wayb.) and you can get the library for Python, 
which is free, and play with it as you like.  To get commas for a linear 
temperament as ratios,

 >>> map(temper.getRatio, temper.Temperament(12,47,
               temper.limit5).getUnisonVectors())
[(6561, 6250)]


                    Graham


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Message: 7115

Date: Sun, 27 Jul 2003 17:54:25

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Carl Lumma wrote:

> For an n-et, one can always think of the generators as 1200 and
> 1200/n.  Then it works the same as for linear temperaments.  Or
> isn't that what you do?

In that case, you're turning an et into a linear temperament in an 
arbitrary way.  If you're consistent between ets, it should be the same 
as taking a fixed proportion of the number of notes to the octave.  For 
comparing equal and linear temperaments it's still arbitrary, and will 
make ets look more complicated then they are.

>>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.  At that n, the complexity would be expected to match the
> heuristic (for linear temperaments) and geometric complexity (for
> everything).  But when n gets smaller we need a way to quantify
> what happens.  It was probably wrong of me to suggest this be some
> sort of temperament complexity.  And I'm wondering if intervals
> achieved through use of the "wolf" break consistency, or whatever
> Gene calls it for regular temperaments.

You can conclude whatever you like.  What Fokker block?  Why bring the 
heuristic into it?  Geometric complexity might work, but I don't 
understand it.

>>For an equal temperament, n can only ever take one value, and
>>n/i=1.
> 
> How do you figure the number of consonant intervals equals the
> number of notes in the et?

I thought consonances were chords.  Otherwise, why distinguish otonal 
and utonal?  It makes more sense for planar temperaments anyway. 
A-B-C-G has all the 5-limit consonances, but no triads.  This can't 
happen with linear temperaments.

>>Taking 5-limit JI as the canonical planar scale, C-E-G gives 1
>>more otonality than utonality.  And C-D-E-G-G#-B has 2 more.
> 
> Do such blocks necessarily involve unison vectors that are larger
> than their smallest 2nd?

I don't know.  Would it always work out for a well formed periodicity 
block?  I'd prefer the complexity didn't depend on the tuning.

>>But you should get the same value for complexity whether you use
>>otonal or utonal.
> 
> Excactly -- which is why they apparently both must be counted
> for planar temperaments.

I thought it was why it didn't matter.


                   Graham


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Message: 7116

Date: Sun, 27 Jul 2003 13:50:11

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> For an n-et, one can always think of the generators as 1200 and
>> 1200/n.  Then it works the same as for linear temperaments.  Or
>> isn't that what you do?
>
>In that case, you're turning an et into a linear temperament in an 
>arbitrary way.  If you're consistent between ets, it should be the
>same as taking a fixed proportion of the number of notes to the
>octave.  For comparing equal and linear temperaments it's still
>arbitrary, and will make ets look more complicated then they are.

So you suggest taking a fixed proportion of the number of notes to
the octave?  What does that mean, and how does one do it?

>>>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.  At that n, the complexity would be expected to match the
>>heuristic (for linear temperaments) and geometric complexity (for
>>everything).  But when n gets smaller we need a way to quantify
>>what happens.  It was probably wrong of me to suggest this be some
>>sort of temperament complexity.  And I'm wondering if intervals
>>achieved through use of the "wolf" break consistency, or whatever
>>Gene calls it for regular temperaments.
>
>You can conclude whatever you like.  What Fokker block?

Any temperament can be viewed in terms of Fokker blocks.

>Why bring the heuristic into it?

Because it tells you the complexity of the temperament.

>Geometric complexity might work, but I don't understand it.

That makes two of us.

>>>For an equal temperament, n can only ever take one value, and
>>>n/i=1.
>> 
>> How do you figure the number of consonant intervals equals the
>> number of notes in the et?
>
>I thought consonances were chords.  Otherwise, why distinguish otonal 
>and utonal?

I was asking about the o/utonal distinction re. your complexity.
I'm suggesting looking at dyads.

>>>Taking 5-limit JI as the canonical planar scale, C-E-G gives 1
>>>more otonality than utonality.  And C-D-E-G-G#-B has 2 more.
>> 
>> Do such blocks necessarily involve unison vectors that are larger
>> than their smallest 2nd?
>
>I don't know.  Would it always work out for a well formed periodicity 
>block?  I'd prefer the complexity didn't depend on the tuning.

But tempering adds intervals/notes, which is how you lower complexity.
So the tuning has to matter.

>>>But you should get the same value for complexity whether you use
>>>otonal or utonal.
>> 
>> Excactly -- which is why they apparently both must be counted
>> for planar temperaments.

So that's important result number 1 from this little exchange.

-Carl


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Message: 7117

Date: Mon, 28 Jul 2003 17:59:32

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Gene Ward Smith wrote:

> Two words: geometric complextity.

Yes, and if you follow back throught the thread you'll see that neither 
Carl nor I understand it.  And explanation would be more useful than 
repeated invocation of the name.


                     Graham


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Message: 7118

Date: Mon, 28 Jul 2003 10:01:52

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> Two words: geometric complextity.
>
>Yes, and if you follow back throught the thread you'll see that neither 
>Carl nor I understand it.  And explanation would be more useful than 
>repeated invocation of the name.

In fairness Gene has tried to explain it.

Paul was working on adapting the heuristic.  You use the heuristic
of both commas, plus the angle between them ("straightness")...

-Carl


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Message: 7119

Date: Mon, 28 Jul 2003 10:39:20

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

Carl Lumma wrote:

> So you suggest taking a fixed proportion of the number of notes to
> the octave?  What does that mean, and how does one do it?

I don't suggest anything, I only know the problems.  If you're counting 
steps upwards from the 1/1, then a 4:5:6 chord will be the same size as 
a 2:3 interval.  For a consistent et of n notes to the octave, that 
means the number of steps you need for a complete triad is the nearest 
integer to 0.585*n.  For an inconsistent et it won't be far off.  So for 
comparing one et to another, you get the same result as comparing the 
number of steps.  For comparing equal with linear temperaments, you get 
7-equal having the same complexity as meantone, which is underestimating 
7-equal.

> Any temperament can be viewed in terms of Fokker blocks.

Usually an infinite number of them!

>>Why bring the heuristic into it?
> 
> Because it tells you the complexity of the temperament.

It gives you an estimate of something you can calculate exactly.  At 
least, the heuristic I'm thinking of does, but that gives accuracy not 
complexity.

> I was asking about the o/utonal distinction re. your complexity.
> I'm suggesting looking at dyads.

For a linear temperament, the complexity of a chord is always the same 
as its most complex interval.  So it doesn't matter which you look at. 
For planar temperaments that needn't be the case, but perhaps it will be 
for sensibly constructed blocks.

> But tempering adds intervals/notes, which is how you lower complexity.
> So the tuning has to matter.

I noticed some mention of wolves before, so perhaps this is it.  It's 
assumed we're dealing with regular temperaments, where the tempering of 
each interval is always the same.  So even if the tuning happens to give 
you some alternative approximations you ignore them.  Hence the 5:1 in 
meantone is always four fifths, even with a Pythagorean tuning where a 
schismic approximation would be closer.  If you wanted that, you should 
have asked for schismic.

In meantone, a fifth is 1 generator step, a major third is 4 generators 
and a minor third is 3 generators.  If that's all you need to calculate 
the complexity, you don't need the tuning.  Most of the complexity 
measures for linear temperaments work like this.  The exception is the 
"smallest MOS" one, because you need some idea of the size of the 
generator to know what MOSs are valid.  That's a practical problem for a 
search algorithm, because it means you have to optimize the tuning for 
any given temperament before you can reject it for being too complex. 
(Then again, for temperaments defined by a pair of ets, you can get a 
list of MOS sizes by working backwards on the scale tree, and this will 
always be sufficient if the seed ets are consistent.)

Because a search for planar temperaments is going to be harder, it would 
be nice if we could still calculate the complexity independent of 
tuning.  But perhaps planar temperaments generated from equal 
temperaments can always be assigned well formed Fokker blocks without 
worrying about the tuning.


                    Graham


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Message: 7121

Date: Mon, 28 Jul 2003 13:24:59

Subject: Re: Creating a Temperment /Comma

From: Gene Ward Smith

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

 Because a search for planar temperaments is going to be harder, it 
would 
> be nice if we could still calculate the complexity independent of 
> tuning.  

Two words: geometric complextity.


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Message: 7123

Date: Mon, 28 Jul 2003 10:57:06

Subject: Re: Creating a Temperment /Comma

From: Carl Lumma

>> Any temperament can be viewed in terms of Fokker blocks.
>
>Usually an infinite number of them!

That's why blocks was plural!

>>>Why bring the heuristic into it?
>> 
>> Because it tells you the complexity of the temperament.
>
>It gives you an estimate of something you can calculate exactly.  At 
>least, the heuristic I'm thinking of does, but that gives accuracy
>not complexity.

There's an error heuristic |n-d|/d*log(d), and a complexity heuristic,
log(d).

>> I was asking about the o/utonal distinction re. your complexity.
>> I'm suggesting looking at dyads.
>
>For a linear temperament, the complexity of a chord is always the same 
>as its most complex interval.  So it doesn't matter which you look at.

But I'm suggesting counting the number of available dyads per number
of notes.

>>But tempering adds intervals/notes, which is how you lower complexity.
>>So the tuning has to matter.
>
>I noticed some mention of wolves before, so perhaps this is it.  It's 
>assumed we're dealing with regular temperaments, where the tempering
>of each interval is always the same.  So even if the tuning happens to
>give you some alternative approximations you ignore them.  Hence the
>5:1 in meantone is always four fifths, even with a Pythagorean tuning
>where a schismic approximation would be closer.  If you wanted that,
>you should have asked for schismic.

Right.  Gene did a search for scales where the wolf was considered to
be another comma.  Or something.  I'm trying to understand how this
fits into the temperament stuff.  His results looked like:

>>Blackwood[10]
>>[0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
>>
>>bad 1662.988586 comp 10.25428060 rms 15.81535241
>>graham 5 scale size 10 ratio 2.000000

>In meantone, a fifth is 1 generator step, a major third is 4 generators 
>and a minor third is 3 generators.  If that's all you need to calculate 
>the complexity, you don't need the tuning.

Ok, but it implies some kind of non-JI tuning.  Maybe I should have
said "ignoring commas increases intervals/notes".

>Most of the complexity measures for linear temperaments work like
>this.  The exception is the "smallest MOS" one,

That doesn't sound like a very good one.

Maybe Gene can prod us to figure out geometric complexity.

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.] 


-Carl


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Message: 7124

Date: Mon, 28 Jul 2003 19:27:23

Subject: Re: Creating a Temperment /Comma

From: Graham Breed

paulhjelmstad wrote:

> This site is cool. I've been playing with it. However, what do I 
> enter (as an example) in "Temperments from Unison Vectors"?

A list of commas that you want tempered out.  For example, 81:80, 
225:224, 1025:1024, 126:125, 50:49 and 64:63 should give some 7-limit 
temperaments.  It's liberal about the format, so you can paste this 
whole message in.


                    Graham


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