Tuning-Math Digests messages 9600 - 9624

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Message: 9600

Date: Sat, 31 Jan 2004 06:56:00

Subject: Re: Nonoctave scales in your livingroom

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> If you don't want to go to the bother of obtaining a musical
> instrument based on 3^(1/13), you might try the temperament with TM
> basis {3125/3087, 6561/6125}. If you take pure tritaves, you get a
> generator of 3^(1/19), or 100.103 cents. If you object that after
> twelve generator steps you get to a pretty good octave of 1201.235
> cents, my reply to you is that 41-et has octaves also. Just ignore 
it.
> Pretend it isn't there, and see what happens.

exactly.


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Message: 9602

Date: Sat, 31 Jan 2004 07:17:27

Subject: Re: 60 for Dave

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >So that he could understand Gene's badness and my linear badness
> >> >in the same form, and propose a compromise.
> >> 
> >> Ah.  Is yours the one from the Attn: Gene post?
> >
> >No, it was the toy "Hermanic" example.
> 
> This, I guess:
> 
> "I thought I'd cull the list of 114 by applying a more stringent
> cutoff of 1.355*comp + error < 10.71. This is an arbitrary choice
> among the linear functions of complexity and error that could be
> chosen"
> 
> You don't say what kind of comp and error you're using.

Copied from Gene's "114".

> >> That's good to know, but the above is just my value judgement, 
and
> >> as you point out log-flat badness frees us from those, in a 
sense.
> >
> >But it results in an infinite number of temperaments, or none at 
all, 
> >depending on what level of badness you use as your cutoff.
> 
> ...as I was trying to complain recently, when I said I'd be a lot
> more impressed if it didn't need cutoffs.

Well you always have at least one cutoff -- namely, the maximum 
allowed value of the badness function itself -- and of the various 
badness functions that have been proposed, a few (including log-flat) 
require one or two additional cutoffs to yield a finite list. A 
straight line doesn't.


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Message: 9603

Date: Sat, 31 Jan 2004 07:23:19

Subject: Re: Crunch algorithm

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > This is similar, if not identical, to Viggo Brun's algorithm that 
> > Kraig is always referring to . . . 
> 
> I'm not so sure--I think possibly Brun's algorithm simply sets out 
to 
> find integer relations, not unimodular matricies. I've never seem 
an 
> exposition of it, just references to it, and don't know how closely 
> what people have said about in tuning connections matches what Brun 
> actually did, which is described as an integer relation algorthm or 
> something along the lines of Jacbobi-Perron when I read about it 
> elsewhere.

Brun may have done more than one thing. Please make it a habit to 
check that big tuning biblography. You'd find this:

http://www.anaphoria.com/viggo0.PDF - Ok *

This language is pretty close to English, but I bet you can just look 
at the numbers and see what's going on.

> However, what I did is identical to the Erv Wilson method--
> he *is* using it to find unimodular matricies, and then inverting 
> them.

Inverting them, huh? Look, he goes all the way out to Orwell. Kraig 
has insisted this is the best method, but I doubt any algorithm could 
perform a three-term integer relation search and do as well as brute 
force for log-flat or whatever badness (except maybe Ferguson-
Forcade?) . . . We should ask John Chalmers if he knows the 
particular Brun reference or if Erv just consulted Mandelbaum.

> > See Mandelbaum's book for a full 
> > exposition . . .
> 
> You'll have to do better than that--is this Joel Mandelbaum,

Yes.

>and what book?

His only:

Mandelbaum, M. Joel. Multiple Division of the Octave and the Tonal 
Resources of the 19-Tone Equal Temperament. PhD Thesis, University of 
Indiana, 1961, 460 pages. University Microfilms, Ann Arbor MI, 1961.


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Message: 9604

Date: Sat, 31 Jan 2004 00:11:40

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

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

> >> But not a chain of one and only one generator.
> >
> >Huh? How not??
> 
> Because he doesn't temper, the generator varies in size
> depending on where you are in the map. 

That's what I meant by an undistributed commatic unison vector. But 
there's still one and only one chain, in contradistinction with 
pajara, augmented, diminished, ennealimmal, etc. . . . which was my 
point.

> >When he does talk about some of these as temperaments (or the 
related 
> >just intonation structures with an undistributed commatic unison 
> >vector), though, they're all single-chain.
> 
> ..of a particular generator in scale steps, not interval size.

Yes.


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Message: 9605

Date: Sat, 31 Jan 2004 08:59:44

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> Yes. I like that idea too. But by "a wide swath" don't you mean one
> that it's easy to put a simple smooth curve thru?

Right.

> And you must have
> some general idea of which way this intergalactic moat must curve.

In the 5-limit linear case, it would be really easy to do this if we 
didn't want to go out to the complexity of schismic and 
kleismic/hanson (the only argument that would arise would be whether 
the father and beep couple should be in or out, leading to a grand 
total of 11 or 9 5-limit LTs). Unfortunately, the low error of 
schismic has proved tantalizing enough for a few musicians to 
construct instruments capable of playing the large extended scales 
that its approximations require. If we consider such complexity 
justifiable, it seems we should be interested in 15 to 17 5-limit 
LTs, or 17 to 19 if we include father and beep. The couple residing 
in "the middle of the road" is 2187;2048 and 3126;2916. With Herman, 
we could split the difference and select only the better of the pair, 
2187;2048 (Dave, have you *heard* Blackwood's 21-equal suite?) . . . 
I don't think anyone's talked about the 20480;19683 system. But if 
schismic, and certainly if semisixths, is not too complex to be a 
useful alternative to strict JI, why shouldn't this system merit some 
attention from musicians too? I don't think near-JI triads sound 
enough better than chords in this system (which are purer than those 
of augmented, porcupine, or diminished) to merit a much higher 
allowed complexity to generate them linearly.

A problem with our plan to have versions of these badness curves for 
sets of temperaments of different dimensions is that moving to a 
higher-dimensional tuning system would theoretically increase the 
badness by an infinite factor. But in practice you never use an 
infinite swath of the lattice so eventually any complex enough 5-
limit linear temperament becomes indistinguishable from a planar 
system.

I'm speaking like you now, Dave! ;)


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Message: 9606

Date: Sat, 31 Jan 2004 00:19:24

Subject: Re: 114 7-limit temperaments

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> What I remember we gave you a hard time about, was not that 
linear
> >> temperaments are 2-dimensional without octave-equivalence, but 
that
> >> you wanted to call them "planar" (which would have been too
> >> confusing a departure from the historical usage). We 
wanted "Linear
> >> temperament" to be the constant name of the musical object which
> >> remains essentially the same while its mathematical
> >> models vary in dimensionality.
> >
> >Unfortunately for us, 'linear temperament' has probably never 
> >referred to a multiple-chains-per-octave system (like pajara, 
> >diminished, augmented, ennealimmal . . .) before we started using 
it 
> >that way, and some of the original users of the term (say, Erv 
> >Wilson) might be rather upset with this slight generalization.
> 
> I can't remember Erv ever using the term,

How about the first line of

http://www.anaphoria.com/xen2.PDF - Ok *

?


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Message: 9608

Date: Sat, 31 Jan 2004 01:31:55

Subject: kleismic v. hanson (was Re: 60 for Dave)

From: Carl Lumma

>In the 5-limit linear case, it would be really easy to do this if we 
>didn't want to go out to the complexity of schismic and 
>kleismic/hanson

While I think it would be nice to name this after Larry Hanson (and
I certainly agreeable to the idea), my preference is to keep kleismic,
since it tells the name of the comma involved, and has a fairly-well
established body of use.  What say everybody?

-Carl


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Message: 9609

Date: Sat, 31 Jan 2004 09:42:58

Subject: the choice of wedgie-norm greatly impacts miracle's ranking

From: Paul Erlich

This time I'll try L_1 (multimonzo interpretation?) instead of 
L_infinity (multival interpretation?) to get complexity from the 
wedgie. Let's see how it affects the rankings -- we don't need to 
worry about scaling because Gene's badness measure is 
multiplicative . . .

The top 10 get re-ordered as follows, though this is probably not the 
new top 10 overall . . .

1.
> 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

39.8287 -> bad = 57.7058

2.
> 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

11.7652 -> bad = 235.1092

3.
> 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

21.1019 --> bad = 280.9843

4.
> 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

7.9560 -> bad = 301.9952 

5.
> 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

15.5360 -> bad = 308.1642

6.
> 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

4.7295 -> bad = 317.0935

7.
> 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

10.4021 -> bad = 336.1437 

8.
> 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

19.9797 -> bad = 377.6573

9.
> 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

20.2918 --> bad = 375.8947

10.
> 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

8.3046 -> bad = 404.8933


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Message: 9610

Date: Sat, 31 Jan 2004 00:37:28

Subject: pelogic and kleismic/hanson

From: Paul Erlich

See

http://www.anaphoria.com/keygrid.PDF - Ok *

page 7 seems to be using some pelog terminology; anyone familiar with 
it?

page 10 provides further support for my proposal of "hanson" as a 
name for kleismic (i.e., dave keenan's chain-of-minor-thirds scale)


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Message: 9611

Date: Sat, 31 Jan 2004 09:46:10

Subject: kleismic v. hanson (was Re: 60 for Dave)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >In the 5-limit linear case, it would be really easy to do this if 
we 
> >didn't want to go out to the complexity of schismic and 
> >kleismic/hanson
> 
> While I think it would be nice to name this after Larry Hanson (and
> I certainly agreeable to the idea), my preference is to keep 
kleismic,
> since it tells the name of the comma involved,

A rare feature, and Pythagorean doesn't eat the Pythagorean comma, 
for example . . .

> and has a fairly-well
> established body of use.  What say everybody?
> 
> -Carl

Fairly-well established body of use?


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Message: 9612

Date: Sat, 31 Jan 2004 02:09:07

Subject: Re: kleismic v. hanson

From: Carl Lumma

>> and has a fairly-well
>> established body of use.  What say everybody?
>
>Fairly-well established body of use?

Yes.

-Carl



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Message: 9613

Date: Sun, 01 Feb 2004 05:30:04

Subject: Re: The true top 32 in log-flat?

From: Carl Lumma

>There's something VERY CREEPY about my complexity values.

Wow dude.  What sort of DES are these?  Not the smallest
apparently.

-Carl


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Message: 9614

Date: Sun, 01 Feb 2004 14:13:25

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>> You're using temperaments to construct scales, aren't you?
>
>Not me, for the most part. I think the non-keyboard composer is 
>simply being ignored in these discussions, and I think I'll stand
>up for him.

How *are* you constructing scales, and what does it have to do
with keyboards?

-Carl


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Message: 9615

Date: Sun, 01 Feb 2004 21:28:52

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

At 09:16 PM 2/1/2004, you wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> >> >> > Such distinctions may be important for *scales*, but for 
>> >> >> > temperaments, I'm perfectly happy not to have to worry about
>> >> >> > them. Any reasons I shouldn't be?
>> >> >> 
>> >> >> You're using temperaments to construct scales, aren't you?
>> >> >
>> >> >Not necessarily -- they can be used directly to construct 
>music, 
>> >> >mapped say to a MicroZone or a Z-Board.
>> >> >
>> >> >http://www.starrlabs.com/keyboards.html *
>> >> 
>> >> ???  Doing so creates a scale.
>> >
>> >A 108-tone scale?
>> 
>> "Scale" is a term with a definition.  I was simply using it.  You
>> meant (and thought I meant?) "diatonic", or "diatonic scale",
>> maybe.
>
>Did you mean a 108-tone scale?

Yes.  -C.


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Message: 9616

Date: Sun, 01 Feb 2004 05:43:05

Subject: Re: The true top 32 in log-flat?

From: Carl Lumma

>> > TOP generators [1201.698520, 504.1341314]

So how are these generators being chosen?  Hermite?  I confess
I don't know how to 'refactor' a generator basis.

-Carl


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Message: 9617

Date: Sun, 01 Feb 2004 14:51:02

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>> I don't think that's quite what Partch says. Manuel, at least, has 
>> always insisted that simpler ratios need to be tuned more accurately, 
>> and harmonic entropy and all the other discordance functions I've 
>> seen show that the increase in discordance for a given amount of 
>> mistuning is greatest for the simplest intervals.
>
>Did you ever track down what Partch said?

Observation One: The extent and intensity of the influence of a
magnet is in inverse proportion to its ratio to 1.

"To be taken in conjunction with the following"

Observation Two: The intensity of the urge for resolution is in
direct proportion to the proximity of the temporarily magnetized
tone to the magnet.

>It also shows that, if all intervals are equally mistuned, the more 
>complex ones will have the highest entropy.

?  The more complex ones already have the highest entropy.  You mean
they gain the most entropy from the mistuning?  I think Paul's saying
the entropy gain is about constant per mistuning of either complex
or simple putative ratios.

>Once you've established that the 9-limit intervals are playable and 
>audible, it may make sense to weight the simple ones higher because
>you expect to use them more.  You could even generate the weights 
>statistically from the score.

I was thinking about this last night before I passed out.  If you
tally the number of each dyad at every beat in a piece of music and
average, I think you'd find the most common dyads are octaves, to be
followed by fifths and so on.  Thus if consonance really *does*
deteriorate at the same rate for all ratios as Paul claims, one
would place less mistuning on the simple ratios because they occur
more often.  This is, I believe, what TOP does.

-Carl


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Message: 9618

Date: Sun, 01 Feb 2004 13:52:34

Subject: Re: 7-limit horagrams

From: Carl Lumma

Beautiful!  I take it the green lines are proper scales?

-C.


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Message: 9619

Date: Sun, 01 Feb 2004 00:16:14

Subject: kleismic v. hanson (was Re: 60 for Dave)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >In the 5-limit linear case, it would be really easy to do this if we 
> >didn't want to go out to the complexity of schismic and 
> >kleismic/hanson
> 
> While I think it would be nice to name this after Larry Hanson (and
> I certainly agreeable to the idea), my preference is to keep kleismic,
> since it tells the name of the comma involved, and has a fairly-well
> established body of use.  What say everybody?
> 
> -Carl

I have to agree with Carl. While Hanson may well be a perfectly
appropriate name for it, when a name has been as extensively used as
"kleismic" (even if only in these archives), you'd not only need to
show what's right about the new name, but also what's wrong about the
old name.


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Message: 9620

Date: Sun, 01 Feb 2004 17:15:20

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Graham Breed

Herman:
>>Schismic and kleismic/hanson start being useful (barely) around 12 
> 
> notes,
> 
>>but the tiny size of schismic steps beyond 12 notes is a drawback 
> 
> until you
> 
>>get to around 41 notes when the steps are a bit more evenly spaced.

Paul E:
> Others may feel differently. Schismic-17 is a favorite of Wilson and 
> others and closely resembles the medieval Arabic system; Helmholtz 
> and Groven used 24 and 36 notes, respectively. Justin White seemed to 
> be most interested in the 29-note version.

I played a lot with the 29 note scale, and it worked fine.  It's nice to 
have some small intervals to throw in when you want them.  It's a 
serious contender because it's still based on fifths, so it's familiar 
and can work with simple adaptations of regular notation.  But that also 
makes it easy to find and so overrated.

It should be included in a 9-limit, or weighted 7-limit, list anyway.


                 Graham


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Message: 9622

Date: Sun, 01 Feb 2004 18:16:40

Subject: Re: 114 7-limit temperaments

From: Graham Breed

Paul Erlich wrote:

> I don't think that's quite what Partch says. Manuel, at least, has 
> always insisted that simpler ratios need to be tuned more accurately, 
> and harmonic entropy and all the other discordance functions I've 
> seen show that the increase in discordance for a given amount of 
> mistuning is greatest for the simplest intervals.

Did you ever track down what Partch said?

Harmonic entropy can obviously be used to prove whatever you like.  It 
also shows that the troughs get narrower the more complex the limit, so 
it takes a smaller mistuning before the putative ratio becomes irrelevant.

It also shows that, if all intervals are equally mistuned, the more 
complex ones will have the highest entropy.  So they're the ones for 
which the mistuning is most problematic, and where you should start for 
optimization.

> Such distinctions may be important for *scales*, but for 
> temperaments, I'm perfectly happy not to have to worry about them. 
> Any reasons I shouldn't be?

You're using temperaments to construct scales, aren't you?  If you don't 
want more than 18 notes in your scale, miracle is a contender in the 
7-limit but not the 9-limit.  And if you don't want errors more than 6 
cents, you can use meantone in the 7-limit but not the 9-limit.  There's 
no point in using intervals that are uselessly complex or inaccurate so 
you need to know whether you want the wider 9-limit when choosing the 
temperament.

> Tenney weighting can be conceived of in other ways than you're 
> conceiving of it. For example, if you're looking at 13-limit, it 
> suffices to minimize the maximum weighted error of {13:8, 13:9, 
> 13:10, 13:11, 13:12, 14:13} or any such lattice-spanning set of 
> intervals. Here the weights are all very close (13:8 gets 1.12 times 
> the weight of 14:13), *all* the ratios are ratios of 13 so simpler 
> intervals are not directly weighted *at all*, and yet the TOP result 
> will still be the same as if you just used the primes. I think TOP is 
> far more robust than you're giving it credit for.

It's really an average over all odd-limit minimaxes.  And the higher you 
get probably the less difference it makes -- but then the harder the 
consonances will be to hear anyway.  For the special case of 7 vs 9 
limit, which is the most important, it seems to make quite a difference.

Once you've established that the 9-limit intervals are playable and 
audible, it may make sense to weight the simple ones higher because you 
expect to use them more.  You could even generate the weights 
statistically from the score.  But there are usually so few usable 
temperaments in any situation, you may as well consider each one 
individually and subjectively.

Oh, yes, I think the 9-limit calculation can be done by giving 3 a 
weight of a half.  That places 9 on an equal footing with 5 and 7, and I 
think it works better than vaguely talking about the number of 
consonances.  After all, how do you share a comma between 3:2 and 9:8? 
I still don't know how the 15-limit would work.

I'm expecting the limit of this calculation as the odd limit tends to 
infinity will be the same as this Kees metric.  And as the integer limit 
goes to infinity, it'll probably give the Tenney metric.  But as the 
integers don't get much beyond 10, infinity isn't really an important 
consideration.

Not that it does much harm either, because the minimax always depends on 
the most complex intervals, which will have roughly equal weighting. 
The same as octave specific metrics give roughly the same results as 
odd-limit style octave equivalent ones if you allow for octave stretching.


                      Graham


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Message: 9623

Date: Sun, 01 Feb 2004 02:47:52

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:

> So the complexity of kleismic in a musically useful sense isn't 
really
> comparable to schismic; this is one thing that the horagrams are 
useful
> for.

The horagrams assume distributionally even, octave-repeating scales; 
our complexity measures don't.


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