Tuning-Math Digests messages 8328 - 8352

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Message: 8328

Date: Mon, 17 Nov 2003 00:29:22

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> If we fix a prime limit, we have duality between multivals and 
> multimonzos; in the 7-limit, a bival and a bimonzo can be identified.
> In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals, can be 
> identified, as can trivals with trimonzos. This involves changing the 
> basis of the n-monzos to make them numerically correspond to the
> (pi(p)-n)-vals. My approach to all this has been to swap the 
> n-monzo for the corresponding (pi(p)-n)-val, and use that, but this 
> does require we fix a prime limit.

Don't we always fix the prime limit anyway? Why might this be a problem?


> > e.g. Why is a 5-limit bi-val a monzo?
> 
> The above duality. It isn't, except if you make the identification.

But why is that identification even possible, when, as I understand
it, the bases are incommensurable? There's something I'm not getting here.

Gene, can you please post your code for calculating the wedge-product
for arbitrary dimensions and arbitrary combinations of grades.

And Graham, if yours uses the same ordering of coefficients, please do
the same. (I know it's in the Python code on your website somewhere,
but I'm lazy).

Any accompanying comments will be gratefully received.

For those who haven't followed John Browne's intro, the "grade" of the
multivector is what we're now indicating by the number of nested
brackets and the "bi-" "tri-" etc.

In the Pascal's-triangle of multivector types, dimension increases
downward and grade increases from left to right.


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Message: 8329

Date: Mon, 17 Nov 2003 13:05:18

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>I also 
>found a couple of scale subsets of a 17-tone octave that have this 
>property, and I have a composition in progress in 17-WT that will 
>illustrate one of these using non-5 harmony.

Sweet!  Can't wait to hear this...

>Carl mentioned (in msg. #7670) that:
>> Paul E. has suggested that we only care about collisions if they
>> occur to a consonant interval.  That allows the diatonic scale
>> in 12-equal to pass.
>
>That's a good hypothesis, but I think that there's more to it than 
>that.  Observe that there is a collision in a harmonic minor scale 
>between the augmented 2nd (a dissonance between the 6th and 7th scale 
>degrees) and the minor 3rds in the scale (which are *consonant*), but 
>I wouldn't say that this results in any disorientation.

Hmm...

>Also consider this:  If we were to make each of the 12-ET fifths in a 
>major scale narrower by 0.1 cents (so as to make the augmented 4th 
>slightly different in size from the diminished 5th), then the scale 
>would be CS, even though we would be hard pressed to hear any 
>difference from 12-ET.  So constant structure (taken alone) is not 
>the whole issue.

It's a bit of a red herring, to use tolerance in this way.  Things
like CS can be assumed to operate through a blur filter.  It's always
better to explicitly spec the filter, as I did for some of Rothenberg's
measures, but anyway....

>2) But if there are two intervals in a scale that are *not 
>functionally different* (such as the two 2:3s or 3:4s in our 11-limit 
>hexatonic otonality), but which span different numbers of steps in 
>the scale, then the possibility for functional scale disorientation 
>exists.  Since it would be very difficult to perceive any interval 
>anywhere in the ballpark of 2:3 or 3:4 as having some other identity 
>or functional role, even tempering to make the scale CS would not 
>address the problem of functional scale disorientation.  So we see 
>that consonance and harmonic entropy are involved here, but it is 
>*functional equality* in combination with a *non-CS* condition for 
>two intervals that are the conditions for disorientation.

Can you write a melody in the 11-limit 'scale' that sounds wrong
because of functional scale disorientation?

>To summarize:  Two intervals with *different* functionality that are 
>the same size will not cause disorientation so long as each one spans 
>the proper number of steps in the scale.  It is only when two 
>intervals with the *same* functionality span differing numbers of 
>steps that the problem arises (regardless of whether they are the 
>same size or not).

[I'm quoting this here so I only have to save this message.]

>I'm not familiar with any of the 11-limit music that Prent Rodgers 
>has produced, so I can only make a guess about why it works.  Suppose 
>that I happen to write a composition using a major scale with the 6th 
>degree omitted.  Is it hexatonic or heptatonic?  I think that we 
>would hear it as heptatonic (due in no small part to the fact that we 
>are so heptatonic-oriented, but on the other hand, I don't know how 
>someone coming from a culture that uses only a pentatonic scale would 
>interpret it).  So I think that it's possible to avoid functional 
>scale disorientation with an 11-limit otonal scale by interpreting it 
>as an incomplete heptatonic scale.

Of all the scales that might sound like a diatonic scale, I think
the 11-limit otonal scale is probably least.  Anyway, you should
definitely download some of Prent's music, even though I don't think
it's a very good example of melodic writing with harmonic series
segments (Prent's music isn't very melodic) -- Denny Genovese or
Jules Siegel have provided much better examples (though not
downloadable at this point).

>I also find that the dodecatonic scale has more tones than I would 
>like to see in anything that I would consider a scale.

We agree on that.

-Carl


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Message: 8330

Date: Mon, 17 Nov 2003 09:44:01

Subject: Re: Vals?

From: monz

hi Gene and Graham (and probably paul too)

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

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

> > I don't know if anybody's following this, but I think
> > the multi-bra notation makes it clearer than anything
> > we've had before.
> 
> I think you and I are probably the only ones without
> headaches, but I do agree. We seem to have passed rapidly
> from disappointing incomprehension to far more
> understanding of this stuff than I've learned to expect,
> and I think the notation thing is a big help. I'm going
> to revise my web pages, and I think revised dictionary
> entries are in order.



yes, absolutely.  i totally agree.  would you guys
*please* rewrite them for me?  thanks.



-monz


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Message: 8331

Date: Mon, 17 Nov 2003 13:07:50

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>> ...
>> But incidentally, I'd love a musical example of a hexatonic 11-limit
>> melody where the non-CS "collision" causes a problem with 
>> constructing a musical sequence.
>
>I'd have to compose something to illustrate that and produce a midi 
>or mp3 file that we could listen to.  Then we would have to decide 
>that, if we both agreed that it didn't work, that it was due to the 
>non-CS property and not because I had a bad composing day.

And would it be so much harder than writing all these messages,
full of so much speculation?

-Carl


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Message: 8332

Date: Mon, 17 Nov 2003 22:38:11

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- 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 "Paul Erlich" 
<perlich@a...> 
> > wrote:
> > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" 
<d.keenan@b...> 
> > > > wrote:
> > > > 
> > > > > If we are told that the mapping is for a tET then _which_ 
tET 
> > it is
> > > > > for can be read straight out of the mapping, as the 
coefficient 
> > for
> > > > > the prime 2 (the first coefficient). And the generator is 
> > simply one
> > > > > step of that tET.
> > > > 
> > > > just wondering why you keep saying "tET" -- 'If we are told 
that 
> > the 
> > > > mapping is for a tone equal temperament then . . .' ??
> > > 
> > > I agree it's awkward. Carl objected so vehemently to EDO and I 
> > wanted
> > > to reserve ET for the most general term (including EDOs ED3s 
cETs).
> > > Perhaps this would be a misuse of ET. Do we have some other 
term for
> > > the most general category of 1D temperaments, i.e. any single
> > > generator temperament whether or not it is an integer fraction 
of 
> > any
> > > ratio? I guess "1D-temperament" will do.
> > > 
> > > > actually, > and < fit together and create a X (as in times) !
> > > 
> > > Oops. Well we could interpret that as the matrix-product as 
opposed 
> > to
> > > the scalar-product (dot-product), but I don't know of any 
meaning 
> > for
> > > that in tuning.
> > 
> > the symbol normally indicates the cross-product, which is 
extremely 
> > useful in tuning: for example, if i take the monzo for the 
diaschisma
> > 
> > [-4 4 -1>
> > 
> > and cross it with the (transpose of the?) monzo for the syntonic 
comma
> > 
> > <-11 4 2]
> 
> Should have been [-11 4 2>

no, the point is that you transpose it so that the angle bracket is 
at the beginning.

> > i get the val for the et where they both vanish:
> > 
> > [12    19    28]
> 
> Now you could write <12 19 28]

maybe.

> That's magic! I never knew that! But of course if someone ever said 
it
> before I wouldn't have understood it since I didn't have a clue 
what a
> val was.
> 
> So [-4 4 -1> (x) [-11 4 2> = <12 19 28]

the idea, though, is that the if the second vector has the angle 
bracket at the beginning, you end up with the symbology ><, which 
already looks like a "x".


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Message: 8333

Date: Mon, 17 Nov 2003 22:43:15

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > so how can i tell which one is covariant and which one is 
> > contravariant?
> 
> Which one do you regard as the vectors you start from 
(contravariant 
> vector) and which as linear functions on the space of such vectors 
> (covariant vector?) Obviously, in our case the monzos are the 
> objects, and the vals are the mappings, and not the other way 
around. 
> However, we *can* consider linear mappings of vals, which can be 
> identifified via unique isomorphim with monzos.
> 
> Anyway, we have this:
> 
> monzo = ket = contravariant
> 
> val = bra = covariant

so it's pretty much a matter of convention which ones you consider 
covariant and which ones you consider contravariant, but you're ok as 
long as you keep the two categories straight? a little math wouldn't 
hurt :)


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Message: 8334

Date: Mon, 17 Nov 2003 22:45:18

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > I agree it's awkward. Carl objected so vehemently to EDO and I 
> wanted
> > to reserve ET for the most general term (including EDOs ED3s 
cETs).
> > Perhaps this would be a misuse of ET. Do we have some other term 
for
> > the most general category of 1D temperaments, i.e. any single
> > generator temperament whether or not it is an integer fraction of 
> any
> > ratio? I guess "1D-temperament" will do.
> 
> Not 1D. These are 0-dimensional temperaments, I'm afraid.

if linear temperaments are 2-dimesional as you always stress, why 
would these be 0-dimensional and not 1-dimensional? for example, 
88cET has a single generator of 88 cents . . . seems 1 dimensional to 
me!


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Message: 8335

Date: Mon, 17 Nov 2003 22:56:57

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > so how can i tell which one is covariant and which one is 
> > contravariant?
> 
> Which one do you regard as the vectors you start from 
(contravariant 
> vector) and which as linear functions on the space of such vectors 
> (covariant vector?) Obviously, in our case the monzos are the 
> objects, and the vals are the mappings, and not the other way 
around. 
> However, we *can* consider linear mappings of vals, which can be 
> identifified via unique isomorphim with monzos.
> 
> Anyway, we have this:
> 
> monzo = ket = contravariant
> 
> val = bra = covariant

in quantum mechanics though, a state can be represented as either a 
bra or a ket, depending on the mathematical operation being 
performed . . .

??


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Message: 8336

Date: Mon, 17 Nov 2003 22:58:04

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > right, but i still want to understand it, since it was in my 
> > relativity textbooks . . .
> 
> It's more complicated in relativity. There you have tangent spaces 
> and cotangent spaces *at every point*, which have to connect 
> together, plus you have a non-positive inner product which changes 
> from point to point. We've got it easy and should enjoy ourseves.

aren't you talking about general relativity? i was only taking 
special relativity . . .


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Message: 8337

Date: Mon, 17 Nov 2003 23:06:18

Subject: Re: Vals?

From: Graham Breed

Dave Keenan wrote:

> And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a
> 7-limit trimonzo will have coefficients ordered [[[e357 e572 e723 e235>>>.
> 
> Is this how your software does it too Graham?

The wedgies are stored in a dictionary, indexed by the bases.  So the 
order only becomes important for some display functions.  I order them 
by increasing index.  And everything uses increasing numbers left to 
right.  So it'd be [[[e235 e237 e257 e357>>>.

> But how do you order the coefficents of a 7-limit bimonzo or bimap
> (bival) so it's its own complement???

Gene does it so you reverse the order to do the complement.  But he's 
never given the general case, and I haven't worked it out.  If I could, 
I might be able to go on to write an efficient implementation in C.


                 Graham


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Message: 8338

Date: Mon, 17 Nov 2003 23:34:17

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > If we fix a prime limit, we have duality between multivals and 
> > multimonzos; in the 7-limit, a bival and a bimonzo can be 
identified.
> > In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals, 
can be 
> > identified, as can trivals with trimonzos. This involves changing 
the 
> > basis of the n-monzos to make them numerically correspond to the
> > (pi(p)-n)-vals. My approach to all this has been to swap the 
> > n-monzo for the corresponding (pi(p)-n)-val, and use that, but 
this 
> > does require we fix a prime limit.
> 
> Don't we always fix the prime limit anyway? Why might this be a 
problem?

sometimes you want to use a set of nonconsecutive primes, as you've 
mentioned yourself, dave.


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Message: 8339

Date: Mon, 17 Nov 2003 23:45:13

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> the idea, though, is that the if the second vector has the angle 
> bracket at the beginning, you end up with the symbology ><, which 
> already looks like a "x".

I'm afraid I don't like that at all. It would only work for two
arguments, not 3 or more, and in any case we don't need to use the
cross-product operator since we're using the more general exterior
product (wedge product) ^. And the < ... ] is meant to tell us we're
looking at a map not a monzo. And it is beneficial to know that you
can't wedge maps with monzos. You have to convert them to the same
kind first.


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Message: 8340

Date: Mon, 17 Nov 2003 23:51:12

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
> wrote: 
> > Anyway, we have this:
> > 
> > monzo = ket = contravariant
> > 
> > val = bra = covariant
> 
> in quantum mechanics though, a state can be represented as either a 
> bra or a ket, depending on the mathematical operation being 
> performed . . .
> 
> ??

I have to admit I'm not too concerned if analogies between quantum
mechanics and tuning theory don't pan out. :-)

I'm only concerned with whether (our extension of) the notation makes
the Grassman Algebra clearer for us. And it certainly seems to be
doing so.


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Message: 8341

Date: Mon, 17 Nov 2003 23:54:01

Subject: Re: "does not work in the 11-limit" (was:: Vals?)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:

> > exactly . . . the two champions would have to be the diatonic 
> > pentatonic and heptatonic scales . . .
> > 
> > > If I'm using a pentatonic scale made from a 9-limit otonal 
chord:
> > > 8 : 9 : 10 : 12 : 14 : 16
> > > then I have two intervals each of 2:3 (both pentatonic "4ths") 
> and 
> > > 3:4 (both pentatonic "3rds").
> > 
> > personally, i'm not fond of this as a scale or melodic entity at 
> all -
> > - when i improvise over a dominant ninth chord, simply using its 
> > notes is about the worst way to come up with a melody . . .
> 
> I understand, and I wouldn't have much to say about the harmonic 
> possibilities either.  But I think that we've been spoiled by the 
> harmonic sophistication of the major-minor system to such an extent 
> that it's difficult to appreciate the resources of a simple scale.  
> We would have to immerse ourselves in gamelan music (particularly 
> slendro) to get in the proper frame of mind to be able to even 
begin 
> to create something decent with such limited tonal resources.  
> (Again, we're off on another topic.)

i don't know . . . i mentioned the diatonic pentatonic scale above. 
that's an equally simple scale, isn't it, and yet i could probably 
live a happy life with no other melodic resources. so it seems you 
missed my point entirely.

i realized, since i made my original post, that the "dominant 
pentatonic" is not CS in 12-equal. perhaps that's one source of my 
difficulty?

> Carl mentioned (in msg. #7670) that:
> > Paul E. has suggested that we only care about collisions if they
> > occur to a consonant interval.  That allows the diatonic scale
> > in 12-equal to pass.
> 
> That's a good hypothesis, but I think that there's more to it than 
> that.  Observe that there is a collision in a harmonic minor scale 
> between the augmented 2nd (a dissonance between the 6th and 7th 
scale 
> degrees) and the minor 3rds in the scale (which are *consonant*), 
but 
> I wouldn't say that this results in any disorientation.

the suggestion of mine that carl was referring to . . . did i ever 
make it quite clear? i don't remember :(

> Also consider this:  If we were to make each of the 12-ET fifths in 
a 
> major scale narrower by 0.1 cents (so as to make the augmented 4th 
> slightly different in size from the diminished 5th), then the scale 
> would be CS, even though we would be hard pressed to hear any 
> difference from 12-ET.  So constant structure (taken alone) is not 
> the whole issue.

well yes, that's exactly the point i was trying to make in my 
original post.

> I believe that the potential for _functional scale disorientation_ 
> (if I may attempt to coin a term) is caused by a particular 
> combination of circumstances:
> 
> 1) If there are two *functionally* different intervals (i.e., aug4 
& 
> dim5, or aug2 & min3) in a scale that only *happen* to be the exact 
> same size (because their ratios, which may be defined either as 
> rational or not, happen to be conflated in the tuning in which the 
> scale is being used) then there is *no* potential for functional 
> scale disorientation.
> 
> 2) But if there are two intervals in a scale that are *not 
> functionally different* (such as the two 2:3s or 3:4s in our 11-
limit 
> hexatonic otonality),

why aren't they functionally different? because we don't have a well-
defined sense of hexatonic musical function, while we know all too 
much about the history and theory of the diatonic scale? i don't 
think that the "happen to" above can be defined in any precise or 
perceptually relevant sense -- though it would be nice . . .

> To summarize:  Two intervals with *different* functionality that 
are 
> the same size will not cause disorientation so long as each one 
spans 
> the proper number of steps in the scale.  It is only when two 
> intervals with the *same* functionality span differing numbers of 
> steps

i look forward to a definition of "functionality" . . .

> I'm not familiar with any of the 11-limit music that Prent Rodgers 
> has produced,

please do listen to it as soon as possible . . .

Microtonal Music by Prent Rodgers *

> so I can only make a guess about why it works.  Suppose 
> that I happen to write a composition using a major scale with the 
6th 
> degree omitted.  Is it hexatonic or heptatonic?  I think that we 
> would hear it as heptatonic (due in no small part to the fact that 
we 
> are so heptatonic-oriented, but on the other hand, I don't know how 
> someone coming from a culture that uses only a pentatonic scale 
would 
> interpret it).  So I think that it's possible to avoid functional 
> scale disorientation with an 11-limit otonal scale by interpreting 
it 
> as an incomplete heptatonic scale.

does the composer or algorithm have to so interpret it in some way 
for this to work?


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Message: 8345

Date: Tue, 18 Nov 2003 11:08:23

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>Me too.  I conceived the idea for it over 25 years ago and started 
>working on it in Cakewalk a little over a year ago.  It's in a 9-tone 
>MOS scale structure -- very different from anything else I've ever 
>tried.  It's been very slow going to figure out how to make 
>everything work to my satisfaction.  I have only about 20 bars of 
>music done, but I'm absolutely delighted with it so far.  At this 
>rate, maybe I'll have it done by the end of this decade.  ;-)

Are you writing it in Sagittal?  ;-)

>> Can you write a melody in the 11-limit 'scale' that sounds wrong
>> because of functional scale disorientation?
>
>I actually tried to run a couple of ideas through my head last night 
>about how to do it, and it's not as easy as I thought.  
>Paradoxically, it takes a certain lack of talent or lack or 
>familiarity with microtonal resources to write something 
>intentionally bad enough that it sounds wrong. -- it's only when I'm 
>not trying to do it that I seem to be able to succeed.  Hmmm, perhaps 
>I've stumbled on a new method of composition -- guaranteed to give 
>good results.  ;-)

Remember, we're trying to ignore composition as a factor.  We don't
care if it's bad, just if it has functional scale disorientation.
If we try and try but can't ever hear functional scale disorientation,
it's important -- it means it probably doesn't exist.

-Carl


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Message: 8346

Date: Tue, 18 Nov 2003 19:59:38

Subject: Re: "does not work in the 11-limit"

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:

> ... I think I've convinced myself that I need to take a break 
> from the tuning lists for a while and use the time to do some 
> composing.  That's the way to discover first-hand which 
> techniques and ideas will work (at least for me) and which
> ones won't.  (And then perhaps I'll also save excerpts of
> some of the bad ideas I try, just for illustration when
> discussions like these come up.)


this is interesting to me.

i have one piece which i wrote in 12edo, and later converted
to JI, but i was never really happy with the JI version.

_In A Minute_:
program notes for In A Minute, (c)1993, 1999 by Joe Monzo *

this page is supposed to open with the mp3, but in case
there's a problem, here it is:
http://sonic-arts.org/monzo/inminute/inminute.mp3 - Ok *


i'm expecting that once my software has music composition
capability (which will be within a month), i'll be able
to do a better job of this.

but anyway, even tho i'm not happy with it, i thought it
was interesting to document my "justification" of this piece.



-monz


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Message: 8348

Date: Tue, 18 Nov 2003 13:28:01

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>No, because it's not actually "written",

Are you entering notes from a keyboard?

-C.


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Message: 8349

Date: Tue, 18 Nov 2003 21:46:55

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> > wrote:
> > > Don't we always fix the prime limit anyway? Why might this be a 
> > problem?
> > 
> > sometimes you want to use a set of nonconsecutive primes, as 
you've 
> > mentioned yourself, dave.
> 
> Good point.
> 
> There must be a convenient way of dealing with these. Does it 
actually
> matter if you use non-consecutive primes, as long as you do it
> consistently throughout the calculations. Isn't it really just the
> _dimension_ of the multi-vectors that must be fixed for any given 
set
> of calculations?

of course. it's just that you might not be dealing with a "prime 
limit"!


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