Tuning-Math Digests messages 9351 - 9375

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Message: 9351

Date: Tue, 20 Jan 2004 01:22:34

Subject: Re: Question for Dave Keenan

From: Carl Lumma

>>  The dominant seventh chord is defined
>> as a local minimum of entropy.
>
>Who defined it as such, when, and why?
>
>As far as I know, the only widely accepted definition of the dominant
>seventh chord is a chord containing the diatonic scale degrees V, VII,
>II, IV. It may or may not be a local harmonic entropy minimum
>depending how the scale is tuned.

Ah; I thought you were referring to '4:5:6:7', when in fact you
want bins to restrict the pitches of the chord.  Finding the highest-
entropy chord that fits in the bins makes sense.

>It seems to me that the more
>dissonant it is, the more relief is likely to be felt when it
>"resolves" to a consonance.

I don't personally agree with this, but it is a rather popular
assertion.

>> >There's also the posssibility that the dominant seventh chord
>> >functions best when its harmonic entropy (or maybe only the HE of one
>> >of its dyads) is locally _maximised_.
>> 
>> What would this look like?
>
>Pretty much like a dominant seventh chord in 12-tET.
>
>Using noble-mediants to estimate them, a max entropy minor seventh
>should be around 1002 cents, a max entropy diminished fifth should be
>around 607 cents, and a max entropy minor third should be around 284
>cents.

If you're trying to maximize pairwise entropy  you'll need to consider
the P-5th and M-3rd too.  But tetradic entropy seems more to the
point.

-Carl


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Message: 9352

Date: Tue, 20 Jan 2004 20:32:15

Subject: Re: Octacot?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > As usual, you need only look here:
> > 
> > Yahoo groups: /tuning/database? *
> > method=reportRows&tbl=10&sortBy=6
> > 
> > or here:
> > 
> > Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. *
> 
> And I was supposed to know we were talking 5-limit exactly how?

I didn't say that, Gene. But dicot and tetracot are there on those 
tables, too, and since you brought them up, it didn't seem wholly 
irrelevant to bring up tricot.


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Message: 9353

Date: Tue, 20 Jan 2004 09:59:44

Subject: Re: A potentially informative property of tunings

From: Dave Keenan

Herman,

I quite agree that it would be very useful to know, for any n-limit
temperament, the max number of generators before we obtain a better
approximation for some n-limit consonance, than that given by the
temperament's mapping.

I'd love to see this figure for all our old favourites. The only thing
that bothers me is that I assume it will vary according to which
particular optimum generator we use, and if so, then it isn't entirely
a property of the temperament (i.e. the map).

But otherwise, it works fine for me to say that temperament X has a
_consistency_limit_ of Y generators.


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Message: 9354

Date: Tue, 20 Jan 2004 21:11:31

Subject: Re: Question for Dave Keenan

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > > >There's also the posssibility that the dominant seventh chord
> > > >functions best when its harmonic entropy (or maybe only the HE 
> of one
> > > >of its dyads) is locally _maximised_.
> > > 
> > > What would this look like?
> > 
> > Pretty much like a dominant seventh chord in 12-tET.
> > 
> > Using noble-mediants to estimate them, a max entropy minor seventh
> > should be around 1002 cents, a max entropy diminished fifth should 
> be
> > around 607 cents, and a max entropy minor third should be around 284
> > cents.
> 
> Why look at these intervals and not the major third, etc.?

Well you _could_ try to locally maximise their entropy too, but I
assumed the dominant triad (without the seventh) normally functions as
a consonance.


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Message: 9356

Date: Tue, 20 Jan 2004 21:13:06

Subject: Re: Compton/Erlich temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Gene W Smith <genewardsmith@j...> 
wrote:
> 
> 
> On Thu, 18 Jul 2002 01:22:19 -0700 Carl Lumma <carl@l...> writes:
> 
> > How do we classify the Compton/Erlich scheme of tuning multiple
> > 12-et keyboards 15 cents apart?  Some sort of planar temperament
> > with the following commas?
> > 
> > 531441/524288 (pythagorean comma)
> > 5120/5103 (difference between syntonic comma and 64/63)
> > 
> > Is this right?
> 
> I think it's another system, discussed below. The wedgie you find 
from
> the pyth comma and 5120/5103 gives what we are calling a linear
> temperament. It is [0,12,12,-6,-19,19], and has a TM reduced basis
> <50/49, 3645/3584>. The mapping is
> 
> [[12, 19, 	28, 34], [0, 0, -1, -1]]
> 
> However, the rms optimum is 23.4 cents apart, not 15.
> 
> I think what you want is the linear temperament with wedgie
> [0,12,12,-6,-19,19], TM reduced basis 
> <225/224, 250047/250000> and mapping [[12,19,28,34],[0,0,-1,-1]]. 

This is the same mapping as above. Did you mean for the last term to 
be -2, not -1? I know Waage proposed this system; who's Compton?


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Message: 9357

Date: Tue, 20 Jan 2004 21:19:14

Subject: the case of the disappearing 250047/250000

From: Paul Erlich

From tuning . . .

I was eliminating commas where the greatest common divisor of the 
entries in the monzo was greater than 1, since these are merely 
powers of other commas. But when I moved to 7-limit, I was 
unfortunately still checking only the first three components of the 
monzo. Hence I erroneously eliminated 250047/250000 and, I think, 
1077 other commas under 600 cents where -20<e3<20, -14<e5<14, and -
12<e7<12.

Both this comma and the most complex one on our recent lists were 
mentioned here:

Yahoo groups: /tuning-math/message/6390 *

and 250047/250000 is mentioned in many posts, for example:
Yahoo groups: /tuning-math/message/4542 *

but even earlier here:
Yahoo groups: /tuning-math/message/1213 *

and its earliest mention on this list was here:
Yahoo groups: /tuning-math/message/200 *


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Message: 9358

Date: Tue, 20 Jan 2004 21:19:56

Subject: Attn: Gene 2

From: Paul Erlich

All right, folks . . . I'm not sure if I missed anything important 
since I last posted, but before I catch up . . .

In the 3-limit, there's only one kind of regular TOP temperament: 
equal TOP temperament. For any instance of it, the complexity can be 
assessed by either

() Measuring the Tenney harmonic distance of the commatic unison 
vector

5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988

() Calculating the number of notes per pure octave or 'tritave':

5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
.........TOP tritave = 1901 -> 19.01 notes per pure tritave.

The latter results are precisely the former divided by 2: in 
particular, the base-2 Tenney harmonic distance gives 2 times the 
number of notes per tritave, and the base-3 Tenney harmonic distance 
gives 2 times the number of notes per octave. A funny 'switch' but 
agreement (up to a factor of exactly 2) nonetheless. In some way, 
both of these methods of course have to correspond to the same 
mathematical formula . . .

In the 5-limit, there are both 'linear' and equal TOP temperaments. 
For the 'linear' case, we can use the first method above (Tenney 
harmonic distance) to calculate complexity. For the equal case, two 
commas are involved; if we delete the entries for prime p in the 
monzos for each of the commatic unison vectors and calculate the 
determinant of the remaining 2-by-2 matrix, we get the number of 
notes per tempered p; then we can use the usual TOP formula to get 
tempered p in terms of pure p and thus finally, the number of notes 
per pure p. Note that there was no need to calculate the angle 
or 'straightness' of the commas; change the angles in your lattice 
and the number of notes the commas define remains the same, so angles 
can't really be relevant here. As I understand it, the determinant 
measures *area* not only in Euclidean geometry, but also in 'affine' 
geometry, where angles are left undefined . . . Anyhow, since both of 
these methods could be used to address a 3-limit TOP temperament, in 
5-limit could they be still both be expressible in a single form in a 
general enough framework, say exterior algebra?

In the 7-limit, the two methods give us, respectively, the complexity 
of a 'planar' temperament as a distance, and the cardinality of a 7-
limit equal temperament as a volume. But 7-limit 'linear' 
temperaments get left out in the cold. The appropriate measure would 
seem to have to be an *area* of some sort -- from what I understand 
from exterior algebra, this is the area of the *bivector* formed by 
taking the *wedge product* of any two linearly independent commatic 
unison vectors (barring torsion). If the generalization I referred to 
above is attainable, all three of the 7-limit cases could be 
expressed in a single way. Anyhow, if this is all correct, I want 
details, details, details. The goal, of course, is to produce 
complexity vs. TOP error graphs for 7-limit linear temperaments, 
something I currently don't know how to do. If someone can fill in 
the missing links on the above, preferably showing the rigorous 
collapse to a single formula in the 3-limit and with some intuitive 
guidance on how to visualize the bivector area in affine geometry (or 
whatever), I'd be extremely grateful.


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Message: 9359

Date: Tue, 20 Jan 2004 13:48:18

Subject: Re: Attn: Gene 2

From: Carl Lumma

This is the 3rd version of this post I've recieved, this time with
a different subject.  Which version should I refer to?

-Carl


>All right, folks . . . I'm not sure if I missed anything important 
>since I last posted, but before I catch up . . .
>
>In the 3-limit, there's only one kind of regular TOP temperament: 
>equal TOP temperament. For any instance of it, the complexity can be 
>assessed by either
>
>() Measuring the Tenney harmonic distance of the commatic unison 
>vector
>
>5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
>12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988
>
>() Calculating the number of notes per pure octave or 'tritave':
>
>5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
>.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
>12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
>.........TOP tritave = 1901 -> 19.01 notes per pure tritave.
>
>The latter results are precisely the former divided by 2: in 
>particular, the base-2 Tenney harmonic distance gives 2 times the 
>number of notes per tritave, and the base-3 Tenney harmonic distance 
>gives 2 times the number of notes per octave. A funny 'switch' but 
>agreement (up to a factor of exactly 2) nonetheless. In some way, 
>both of these methods of course have to correspond to the same 
>mathematical formula . . .
>
>In the 5-limit, there are both 'linear' and equal TOP temperaments. 
>For the 'linear' case, we can use the first method above (Tenney 
>harmonic distance) to calculate complexity. For the equal case, two 
>commas are involved; if we delete the entries for prime p in the 
>monzos for each of the commatic unison vectors and calculate the 
>determinant of the remaining 2-by-2 matrix, we get the number of 
>notes per tempered p; then we can use the usual TOP formula to get 
>tempered p in terms of pure p and thus finally, the number of notes 
>per pure p. Note that there was no need to calculate the angle 
>or 'straightness' of the commas; change the angles in your lattice 
>and the number of notes the commas define remains the same, so angles 
>can't really be relevant here. As I understand it, the determinant 
>measures *area* not only in Euclidean geometry, but also in 'affine' 
>geometry, where angles are left undefined . . . Anyhow, since both of 
>these methods could be used to address a 3-limit TOP temperament, in 
>5-limit could they be still both be expressible in a single form in a 
>general enough framework, say exterior algebra?
>
>In the 7-limit, the two methods give us, respectively, the complexity 
>of a 'planar' temperament as a distance, and the cardinality of a 7-
>limit equal temperament as a volume. But 7-limit 'linear' 
>temperaments get left out in the cold. The appropriate measure would 
>seem to have to be an *area* of some sort -- from what I understand 
>from exterior algebra, this is the area of the *bivector* formed by 
>taking the *wedge product* of any two linearly independent commatic 
>unison vectors (barring torsion). If the generalization I referred to 
>above is attainable, all three of the 7-limit cases could be 
>expressed in a single way. Anyhow, if this is all correct, I want 
>details, details, details. The goal, of course, is to produce 
>complexity vs. TOP error graphs for 7-limit linear temperaments, 
>something I currently don't know how to do. If someone can fill in 
>the missing links on the above, preferably showing the rigorous 
>collapse to a single formula in the 3-limit and with some intuitive 
>guidance on how to visualize the bivector area in affine geometry (or 
>whatever), I'd be extremely grateful.
>
>
> 
>
>
>Yahoo! Groups Links
>
>To visit your group on the web, go to:
> Yahoo groups: /tuning-math/ *
>
>To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@xxxxxxxxxxx.xxx
>
>Your use of Yahoo! Groups is subject to:
> Yahoo! Terms of Service *


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Message: 9360

Date: Tue, 20 Jan 2004 22:03:43

Subject: Re: the case of the disappearing 250047/250000

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> From tuning . . .
> 
> I was eliminating commas where the greatest common divisor of the 
> entries in the monzo was greater than 1, since these are merely 
> powers of other commas. But when I moved to 7-limit, I was 
> unfortunately still checking only the first three components of the 
> monzo. Hence I erroneously eliminated 250047/250000 and, I think, 
> 1077 other commas under 600 cents where -20<e3<20, -14<e5<14, and -
> 12<e7<12.

And apparently 391 others under 600 with log(n*d)/log(10)<18, which 
is what the graphs were showing :(

I've uploaded the 3 corrected graphs.


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Message: 9361

Date: Tue, 20 Jan 2004 22:06:15

Subject: Grassmann Algebra question

From: Paul Erlich

Can someone explain equation (7) here:

http://www.maths.utas.edu.au/People/dfs/Papers/GrassmannLinAlgpaper.pd *
f

?

Why do we 'assume' it in accordance with what is written above it?


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Message: 9363

Date: Tue, 20 Jan 2004 22:53:08

Subject: Re: Maple code for ?(x)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> This shows how easy it is to compute, and may inspire anyone who 
wants
> to code it in something other than Maple. Paul was using a log-based
> system for defining harmonic entropy, and that makes more sense, so
> the actual function to do Stieltjes integration with would be ?(2^x)
> or ?(2^(x/1200)), but either way you want a ? function to start out 
with.
> 
> quest := proc(x)
> # Minkowsky question mark function
> local i, j, d, l, s, t;
> l := convert(x, confrac);
> d := nops(l);
> s := l[1];
> for i from 2 to d do
> t := 1;
> for j from 2 to i do
> t := t - l[j] od;
> s := s + (-1)^i * 2^t od;
> s end:

what is nops?


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Message: 9366

Date: Wed, 21 Jan 2004 13:38:47

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>> This list is attractive, but Meantone, Magic, Pajara, maybe
>> Injera to name a few are too low for my taste, if I'm reading
>> these errors right (they're weighted here, I take it).
>> 
>> If you could make this list finite with badness bounds only,
>> I'd be more impressed by claims that log-flat badness is
>> desirable (allows the comparison of ennealimmal with all
>> temperaments in a sense, not just the others on the list, or
>> whatever).
>
>Log flat badness is deliberately designed not to be finite. and it
>seems to me your objection is strange--do you think epimericity
>allows comparison of one comma with another, while a log flat badness
>does not?

What's the rub again?  Within equally-sized complexity bins, log-flat
badness returns roughly the same number of temperaments?  I guess
that makes sense.

>As for meantone, magic and pajara being too low, they are all near tht
>top of the list. It would seem the list is doing exactly what you want
>it to do.

I did say it was attractive...

>You can make the list finite by bounding complexity, which is what
>I've done.
>
>> And I don't see how you figure schismic is less complex than
>> miracle in light of the maps given.
>
>Schismic gets to 3/2 in one generator step, and miracle takes six.

What kind of complexity is this?  Do you always use the same kind?
It seems you happily switch between geometric, weighted map-based,
and 3 other flavors when giving these lists.  Providing a template
at the top of the lists showing units or something for each key
might help your readers.

-Carl


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Message: 9370

Date: Wed, 21 Jan 2004 13:43:27

Subject: Re: Maple code for ?(x)

From: Carl Lumma

>> Obviously, Maple is not going to compute an infinte number of
>> convergents. What it does depends on the setting of Digits.
>
>So we're getting some *approximation* to the ? function, yes? Matlab 
>does continued fractions, but they're strings not vectors, they use 
>an error tolerance to determine when to stop, and they allow negative 
>entries. I've written code to get around the last limitation; the 
>other two shouldn't be that hard . . .

By the way, somebody once told me that maple code can be executed
in Matlab.  Dunno if that's true...

-Carl


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Message: 9373

Date: Wed, 21 Jan 2004 02:15:15

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

This list is attractive, but Meantone, Magic, Pajara, maybe
Injera to name a few are too low for my taste, if I'm reading
these errors right (they're weighted here, I take it).

If you could make this list finite with badness bounds only,
I'd be more impressed by claims that log-flat badness is
desirable (allows the comparison of ennealimmal with all
temperaments in a sense, not just the others on the list, or
whatever).

And I don't see how you figure schismic is less complex than
miracle in light of the maps given.

-Carl

>Number 1 Ennealimmal
>
>[18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
>TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
>TOP generators [133.3373752, 49.02398564]
>bad: 4.918774 comp: 11.628267 err: .036377
>
>
>Number 2 Meantone
>
>[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
>TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>TOP generators [1201.698520, 504.1341314]
>bad: 21.551439 comp: 3.562072 err: 1.698521
>
>
>Number 3 Magic
>
>[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
>TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>TOP generators [1201.276744, 380.7957184]
>bad: 23.327687 comp: 4.274486 err: 1.276744
>
>
>Number 4 Beep
>
>[2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
>TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>TOP generators [1194.642673, 254.8994697]
>bad: 23.664749 comp: 1.292030 err: 14.176105
>
>
>Number 5 Augmented
>
>[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
>TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
>TOP generators [399.9922103, 107.3111730]
>bad: 27.081145 comp: 2.147741 err: 5.870879
>
>
>Number 6 Pajara
>
>[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
>TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>TOP generators [598.4467109, 106.5665459]
>bad: 27.754421 comp: 2.988993 err: 3.106578
>
>
>Number 7 Dominant Seventh
>
>[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
>TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
>TOP generators [1195.228951, 495.8810151]
>bad: 28.744957 comp: 2.454561 err: 4.771049
>
>
>Number 8 Schismic
>
>[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
>TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>TOP generators [1200.760624, 498.1193303]
>bad: 28.818558 comp: 5.618543 err: .912904
>
>
>Number 9 Miracle
>
>[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>TOP generators [1200.631014, 116.7206423]
>bad: 29.119472 comp: 6.793166 err: .631014
>
>
>Number 10 Orwell
>
>[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
>TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
>TOP generators [1199.532657, 271.4936472]
>bad: 30.805067 comp: 5.706260 err: .946061


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Message: 9374

Date: Wed, 21 Jan 2004 21:52:22

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >Number 8 Schismic
> >> >
> >> >[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> >> >TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> >> >TOP generators [1200.760624, 498.1193303]
> >> >bad: 28.818558 comp: 5.618543 err: .912904
> >> >
> >> >
> >> >Number 9 Miracle
> >> >
> >> >[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> >> >TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> >> >TOP generators [1200.631014, 116.7206423]
> >> >bad: 29.119472 comp: 6.793166 err: .631014
> //
> >> And I don't see how you figure schismic is less complex than
> >> miracle in light of the maps given.
> >
> >Probably the shortness of the fifths in the lattice wins it for 
> >schismic . . .
> 
> After I wrote that I reflected a bit on comma complexity vs. map
> complexity.  Comma complexity gives you the number of notes you'd
> have to search to find the comma, on average (Kees points out that
> the symmetry of the lattice allows you to search 1/4 this numeber
> in the 5-limit, or something, but anyway...).  Map complexity is
> the number of notes you need to complete the map *with contiguous
> chains of generators*. 

Thus it will depend on the choice of generators. For so-called linear 
temperaments, this is only made definite by fixing one of them to be 
1/N octaves. For planar and higher-dimensional temperaments, the 
choice is even more arbitrary. Comma complexity, or wedgie complexity 
for higher codimensions, is well-defined, and is (according to Gene) 
the natural generalization of the complexity measures we all agree on 
for the simplest cases.


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