A tuning-math question -- finding 2D temperaments from 4D ratios

Hi tuners.

I'm not 100% sure what the exact mathematical steps should be for solving this but I hope you'll understand my explanation.

I'm thinking of an algorithm which would take a 7-limit factor and find the two approximants for the resulting 2D temperament, including the mapping. For example, let's say that the monzo (or "PSC" as I was calling it earlier) is [-11 2 7 -3]. From this I should find that it's actually the distance between (12/5)^7 and (54/7)^3. Whatever method I choose to compare these intervals, this suggests the approximate size of the generator and says that I use 3 of them for approximating 12/5 and 7 for approximating 54/7. Next, I'll find such octave equivalents or octave inversions of these two which are as low in the harmonic series as possible, which is actually 5/3 and 27/7, respectively. If there were a piece of code doing it, these should be returned among the "output values" for the end user because we're no longer approximating 3/1 and 5/1. Finally, I'll give the mapping for the new approximants, including the octave, which is [(1 0) (2 -3) (1 -7)].

Now the thing is that I know what the results should be for some ratios because of other occasions but I'm not sure how I should do this for an unfamiliar ratio.

Another example is the earlier discussed [10 5 8 -13] which actually compares (12/7)^13 to (12/5)^8. So this time, the target approximants are 5/3 and 7/3 and the full mapping is [(1 0) (2 -13) (2 -8)].

Obviously, if the GCD of the 3-5-7 exponents is not 1, then we have to split the octave -- but that's another story.

If you have an idea how this could be applied to other ratios, I welcome any suggestions.

Thanks in advance.

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:
Whatever method I choose to compare these
> intervals, this suggests the approximate size of the generator and says that
> I use 3 of them for approximating 12/5 and 7 for approximating 54/7.

An easier approach, it seems to me, is to first find the normal val list, which has corresponding generators 2, 3 and 75/56. The last is what you wanted: (12/5)/(75/56)^3 is 1/c and (54/7)/(75/56)^7 = 1/c^2, where c is your comma 703125/702464. If you want a 75/56 generator for a linear temperament, you can start with an et tempering out your comma and try to find one which does not lead to a period which is a fraction of an octave. There's probably a better way than that to do things, but using that method with 140 I quickly found the 19&121 temperament, which fits the bill.

I wrote:

> Finally, I'll give the mapping for the new > approximants, including the octave, which is [(1 0) (2 -3) (1 -7)].

I meant "(-1 7)".

Petr

Gene wrote:

> If you want a 75/56 generator for a linear temperament, you can start with > an et tempering out your comma and
> try to find one which does not lead to a period which is a fraction of an > octave. There's probably a better way than that to do things, but using > that method with 140 I quickly found the 19&121 temperament, which fits > the bill.

What I primarily wanted was to find the two approximated ratios because here they aren't 3/1 and 5/1 as with 5-limit temperaments. So I was trying to answer the question that in this particular case, they are 5/3 and 27/7 and that in the case of [10 5 8 -13], they are 5/3 and 7/3. I just don't know how to do it. Once I know what they are, I can derive the mapping for the corresponding 2D temperaments.

Petr

Hi Petr,

> For example, let's say that the monzo is [-11 2 7 -3].
> From this I should find that it's actually the distance
> between (12/5)^7 and (54/7)^3

Hm, not sure I understand the question. Would the
following method work? Given a 7-limit monzo,

discard the first element (2 exponent)
let p be the largest positive element remaining
let n be the largest (abs) negative element remaining
let m be the last element remaining
let p', n', m' denote corresponding primes for p, n, and m

answer will be of the form (x/p')^p (y/n')^n where

ny/m' - px/m' = m

assuming you sprinkle factors of 2 back in so that
(x/p')^p < (y/n')^n. I think. When m is positive.
There may be a variation when m is negative.

-Carl

Carl wrote:

> discard the first element (2 exponent)
> let p be the largest positive element remaining
> let n be the largest (abs) negative element remaining
> let m be the last element remaining
> let p', n', m' denote corresponding primes for p, n, and m
>
> answer will be of the form (x/p')^p (y/n')^n where
>
> ny/m' - px/m' = m
>
> assuming you sprinkle factors of 2 back in so that
> (x/p')^p < (y/n')^n. I think. When m is positive.
> There may be a variation when m is negative.

Wow, this looks really promissing!
Now the only thing that remains is to find the final steps which tell us that x and y are "12, 54" in the first example and "12, 12" in the second one.
Once we crack this down, we can expect a whole bunch of new 2D tunings waiting to be discovered -- i.e. the target subgroup would be determined by the vanishing interval rather than the other way round ... Heck, this looks crazy. :-D

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:

> Once we crack this down, we can expect a whole bunch of new 2D tunings
> waiting to be discovered -- i.e. the target subgroup would be determined by
> the vanishing interval rather than the other way round ...

I'm not sure I understand you. I was just mentioning 325/324, and I suppose you could say the subgroup 2.3.5.13 is "determined by" it, as those are the only primes which appear in its factorization. If I wanted to find out what a good subgroup would be for 385/384, how do I go about it? Or does this not apply to commas like 385/384, which can be regarded as simply tempering the 7-limit to get 11-limit harmony, which is a pretty typical situation, and which differs from 325/324 only in that for 325/324 the 7 and 11 are simply added in on top, so to speak.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

>Or does this not apply to commas like 385/384, which can be regarded as simply tempering the 7-limit to get 11-limit harmony, which is a pretty typical situation, and which differs from 325/324 only in that for 325/324 the 7 and 11 are simply added in on top, so to speak.

One question is how the spectrum might help:

http://xenharmonic.wikispaces.com/Spectrum+of+a+temperament

For 385/384, you get nothing. For 325/324, you get that 10/9 might be important and that 11 is pretty far down the totem pole. I don't see that it's very helpful, but it's something.

Petr wrote:

> > discard the first element (2 exponent)
> > let p be the largest positive element remaining
> > let n be the largest (abs) negative element remaining
> > let m be the last element remaining
> > let p', n', m' denote corresponding primes for p, n, and m
> > answer will be of the form (x/p')^p (y/n')^n where
> >
> > ny/m' - px/m' = m
> >
> > assuming you sprinkle factors of 2 back in so that
> > (x/p')^p < (y/n')^n. I think. When m is positive.
> > There may be a variation when m is negative.
>
> Wow, this looks really promising!
> Now the only thing that remains is to find the final steps
> which tell us that x and y are "12, 54" in the first example
> and "12, 12" in the second one.
> Once we crack this down, we can expect a whole bunch of
> new 2D tunings waiting to be discovered -- i.e. the target
> subgroup would be determined by the vanishing interval rather
> than the other way round ... Heck, this looks crazy. :-D

Somebody like Graham could probably work out the rest of
the details and fix the sign I got wrong in a few minutes.
Me, it would take the better part of an hour.

-Carl

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> For 385/384, you get nothing. For 325/324, you get that 10/9 might be important and that 11 is pretty far down the totem pole. I don't see that it's very helpful, but it's something.

Here's another approach: for 703125/702464, a minimal temperamental complexity basis for the lattice of pitch classes of the temperament is given by approximations of [75/56, 28/25]. So these, and especially 75/56, get singled out.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@> wrote:
>
> > Once we crack this down, we can expect a whole bunch of new 2D tunings
> > waiting to be discovered -- i.e. the target subgroup would be determined by
> > the vanishing interval rather than the other way round ...
>
> I'm not sure I understand you. I was just mentioning 325/324, and I suppose you could say the subgroup 2.3.5.13 is "determined by" it, as those are the only primes which appear in its factorization.

Okay, I see you're doing something similar except that I was trying to find out what the target approximated ratios were in a 2D temperament rather than what primes to involve in a 3D temperament. Carl has suggested a nice algorithm for doing the job and now I only have to figure out how to find the x and y.
Petr

Hi Petr,

> Okay, I see you're doing something similar except that I was
> trying to find out what the target approximated ratios were in
> a 2D temperament rather than what primes to involve in a
> 3D temperament. Carl has suggested a nice algorithm for doing
> the job and now I only have to figure out how to find
> the x and y.
> Petr

I just tried working through your example

> For example, let's say that the monzo is [-11 2 7 -3].
> From this I should find that it's actually the distance
> between (12/5)^7 and (54/7)^3

> discard the first element (2 exponent)
> let p be the largest positive element remaining
> let n be the largest (abs) negative element remaining
> let m be the last element remaining

p=7, n=3, m=2

> let p', n', m' denote corresponding primes for p, n, and m
> answer will be of the form (x/p')^p (y/n')^n where
> ny/m' - px/m' = m

That'll be 3y/3 - 7x/3 = 2. Wolfram Alpha says x=3, y=9
works. That gives

(3/5)^7 (9/7)^3

Hm, the numerator of the second fraction seems to be
missing a factor of 3 vs. your solution. . . -Carl

Carl wrote:

> Hm, the numerator of the second fraction seems to be
> missing a factor of 3 vs. your solution. . .

Carl, thanks for your time anyway because with a bit of your help, I think I've eventually found the steps. The thing which was stopping me and which you suggested was to treat m separately and differently from p and n. So if I add one more symbol to the ones you assigned and label it "o", then I have [-11o 2m 7p -3n] and I'll break it into two sets of 4 numbers. One will have a zero in place of p and n will be unchanged, the other will have zero in place of n and p will be unchanged, and both o and m will be equal to the unchanged n or p multiplied by an integer of the lowest possible absolute value (although I'm not sure how I would say to a computer to find the lowest(abs) possible multiplier here). This means:
[-11 2 7 -3] = [-14 -7 7 0] + [3 9 0 -3]
Voila, that's exactly what we were looking for. :-)

Similarly, if we start with [10o 5m 8p -13n], we have:
[10 5 8 -13] = [-16 -8 8 0] + [26 13 0 -13]

Now let me tell you a sweet secret. Although I wasn't aware that it could work in this case, it's actually exactly the same procedure I've been using for finding 5-limit 2D temperaments. For example, the version with the lowest(abs) multipliers for amity is:
[9 -13 5] = [39 -13 0] + [-30 0 5]

Wow, I'm so happy that we've nailed it down in the end ... Now I suspect that there are some interesting 2D tunings hidden in these 7-limit factors. I'm only afraid it would take me whole days to get to them because finding the lowest multipliers by hand is terribly slow, or at least when I'm doing it -- but I've done it many times already so I'll do it a few times more -- at least I'll refresh my math skills that way.and then, what comes next ... Music.

Petr

Hi Petr, a few questions...

On Jun 23, 2011, at 2:24 PM, "Petr Pařízek" <petrparizek2000@...>
wrote:

Hi tuners.

I'm not 100% sure what the exact mathematical steps should be for solving
this but I hope you'll understand my explanation.

I'm thinking of an algorithm which would take a 7-limit factor and find the
two approximants for the resulting 2D temperament, including the mapping.

What do you mean by two approximants? The mapping for the two generators?

For example, let's say that the monzo (or "PSC" as I was calling it earlier)

is [-11 2 7 -3]. From this I should find that it's actually the distance
between (12/5)^7 and (54/7)^3.

Why these two intervals, and how did you figure this out?

Whatever method I choose to compare these
intervals, this suggests the approximate size of the generator and says that

I use 3 of them for approximating 12/5 and 7 for approximating 54/7. Next,
I'll find such octave equivalents or octave inversions of these two which
are as low in the harmonic series as possible, which is actually 5/3 and
27/7, respectively. If there were a piece of code doing it, these should be
returned among the "output values" for the end user because we're no longer
approximating 3/1 and 5/1.

How are we no longer approximating 3/1 and 5/1? You mean in this new
subspace the basis vectors are 2/1, 5/3, and 27/7?

-Mike

Hi Mike.

I wrote:

> I'm thinking of an algorithm which would take a 7-limit factor and find the
> two approximants for the resulting 2D temperament, including the mapping.
>
> What do you mean by two approximants? The mapping for the two
> generators?

I mean the ratios we're approximating instead of 3/1 and 5/1.

>
> For example, let's say that the monzo (or "PSC" as I was calling it earlier)
>
> is [-11 2 7 -3]. From this I should find that it's actually the distance
> between (12/5)^7 and (54/7)^3.
>
> Why these two intervals, and how did you figure this out?

Go to Graham's "uv.html" and type in 703125/702464.

>
> Whatever method I choose to compare these
> intervals, this suggests the approximate size of the generator and says that
>
> I use 3 of them for approximating 12/5 and 7 for approximating 54/7. Next,
> I'll find such octave equivalents or octave inversions of these two which
> are as low in the harmonic series as possible, which is actually 5/3 and
> 27/7, respectively. If there were a piece of code doing it, these should be
> returned among the "output values" for the end user because we're no longer
> approximating 3/1 and 5/1.
>
> How are we no longer approximating 3/1 and 5/1? You mean in this new
> subspace the basis vectors are 2/1, 5/3, and 27/7?

Exactly.

Nevertheless, if you read message # 100342, you'll see that I've eventually managed to find the way to do it with some "starting inspiration" from Carl.
I've just tried it a while ago with 4375/4374 and that's why I've started a new thread.

Petr

On Jun 24, 2011, at 2:07 PM, "petrparizek2000" <petrparizek2000@...>
wrote:

>
> How are we no longer approximating 3/1 and 5/1? You mean in this new
> subspace the basis vectors are 2/1, 5/3, and 27/7?

Exactly.

Ok, but I'm still confused. You could say that the basis vectors for 5-limit
JI are 5/3, 25/24, and 16/5 if you want, and you'll still get 5-limit JI.
Likewise you can say the basis vectors for the 3D 225/224 planar temperament
are 2/1, 3/1, and 5/1. It's like saying the basis vectors for meantone are
2/1 and 3/1 vs 25/24 and 128/125. You probably know this.

So in this case you've used Graham's algorithm to get some kind of normal
val list, then you get the 3 normal generators for the 3D temperament. Then
you pick the two that aren't 2/1, and then you remove factors of 2 from
them. And then what happens? You try to express the comma via the two
remaining vectors, and find a way to reduce the dimensionality further by
doing so?

Nevertheless, if you read message # 100342, you'll see that I've eventually
managed to find the way to do it with some "starting inspiration" from Carl.
I've just tried it a while ago with 4375/4374 and that's why I've started a
new thread.

I'm just confused about the point of this algorithm - is it to find a
"canonical" rank 2 temperament from a codimension 1 rank 3 temperament?

-Mike

Mike wrote:

> I'm just confused about the point of this algorithm - is it to find a
> "canonical" rank 2 temperament from a codimension 1 rank 3 temperament?

Although I'm not sure if this is what you meant, I'll at least say it in my own words. The algorithm tries to find the approximated ratios used in a 2D temperament with a given vanishing interval. For 5-limit 2D temperaments, they are 3/1 and 5/1. However, if you want to find a 2D tuning tempering out 703125/702464, then you have to approximate some other ratios. Note that there's a difference in making a 2D temperament by tempering out two 7-limit commas and making a 2D temperament by tempering out one 7-limit comma. In the former case, you can happily use 3 target approximants, those being 3/1, 5/1, 7/1. In the latter, you can only use 2. And this is what I was trying to answer -- what would they be for one comma, what would they be for another comma, and so on.
Petr

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:

> Although I'm not sure if this is what you meant, I'll at least say it in my own words. The algorithm tries to find the approximated ratios used in a 2D temperament with a given vanishing interval. For 5-limit 2D temperaments, they are 3/1 and 5/1. However, if you want to find a 2D tuning tempering out 703125/702464, then you have to approximate some other ratios.

As far as I can tell using generators of small complexity provides a good answer to your question but I'm still not entirely sure I understand what the question is.

On Jun 24, 2011, at 4:14 PM, petrparizek2000 <petrparizek2000@...>
wrote:

Mike wrote:

> I'm just confused about the point of this algorithm - is it to find a
> "canonical" rank 2 temperament from a codimension 1 rank 3 temperament?

Although I'm not sure if this is what you meant, I'll at least say it in my
own words. The algorithm tries to find the approximated ratios used in a 2D
temperament with a given vanishing interval. For 5-limit 2D temperaments,
they are 3/1 and 5/1. However, if you want to find a 2D tuning tempering out
703125/702464, then you have to approximate some other ratios.

Why can't you still treat the generators as 3/1 and 5/1?

Note that there's a difference in making a 2D temperament by tempering out
two 7-limit commas and making a 2D temperament by tempering out one 7-limit
comma.

What do you mean? If you're starting with 7-limit JI, then you need to
temper out two commas to get to rank 2... You obviously already know this,
so I'm misunderstanding something.

In the former case, you can happily use 3 target approximants, those being
3/1, 5/1, 7/1.

You mean the latter case? The former case is 2D, so I don't see how there
are 3 approximants. And are you using the word approximant to mean generator
here?

In the latter, you can only use 2. And this is what I was trying to answer
-- what would they be for one comma, what would they be for another comma,
and so on.

Sorry man, I'm really confused. It looks to me like this is a way to find
good 2D child temperaments from a 3D parent temperament. Am I right?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
However, if you want to find a 2D tuning tempering out
> 703125/702464, then you have to approximate some other ratios.
>
> Why can't you still treat the generators as 3/1 and 5/1?

Because you won't get all of the 7-limit intervals.

Mike wrote:

> Why can't you still treat the generators as 3/1 and 5/1?

First of all, if you temper out one interval within a 3D untempered system, you get a 2D tempered system. That means you started with 3 ratios, one of which you use as the period and the 2 others are the target approximated ratios. So there are one more non-period approximated ratios than there are generators -- and for 5-limit temperaments, for example, they are 3/1 and 5/1, no matter how one particular temperament maps them in terms of periods and generators.

Now, if you want to use target approximated ratios of 3/1 and 5/1 within a 7-limit system and still get a 2D result, you then have to temper out two commas. But this is not what I'm trying to do here. In fact, I'm doing the opposite of what would normally be the goal. To give you an example, in the more "normal" case, I could find a "3-less" 2D temperament by first deciding that the period (or the equivalence interval, to be precise) should be 2/1, then deciding that the target approximated factors should be 5/1 and 7/1, and finally tempering out one interval using these "input data" as my starting point (which obviously offers me commas like 50/49 or 3136/3125 or that sort of thing). So in that case, I first had three requirements: "Use 2/1 as the equivalence interval, use 5/1 as one approximant, use 7/1 as another approximant"; and then I found a 2D temperament based on these three.

However, now let's suppose we have not three but two starting requirements: "Use 2/1 as the equivalence interval, find a 2D tuning which tempers out such and such a comma". If the vanishing interval contains three primes and one of them is two, then the procedure is obvious (for example, when tempering out 3136/3125, 2 generators approximate 5/4 and 5 generators approximate 7/4). However, what if it's something like 225/224, which contains 4 different primes (which would suggest to use 3 target approximants instead of 2, those being 3/1, 5/1, 7/1), and you still want a 2D temperament? Well, you have basically three options to choose from. #1: You can temper out two commas at a time, which makes it perfectly possible to start with 3 target approximated factors instead of 2. #2: You can make a non-octave temperament that would use a tempered 5/4 as the generator and a tempered 4/3 or 5/3 as the period (which is what I've done in my little "Non-octave marvel" study last year). #3: You can spot that 225/224 is (8/7)^1 / (16/15)^2; so you can happily use an octave as the period and a tempered minor second as the generator, two of which make a tempered 8/7. Note that in this third example, we're approximating 16/15 without actually including 3/2 or 5/4. That means that our target approximated odd factors now are 15/1 and 7/1 (I'm stripping the 2 out because that's our period). And this "option #3" is what I'm thinking of here. Of course, if you wish to approximate 3/1 or 5/1, you can add another generator (let's say 5/4) and you end up with marvel. This means you would then need to stack at least two of these "secondary generators" to get a 5-limit triad -- i.e. if the primary generator is 16/15 and the secondary generator is 5/4, then 5/3 is mapped to "1, 2" in terms of their counts. If you favor 5-limit intervals over 7-limit ones, you can also get marvel by using generators of, for example, 3/1 and 5/1. The downside of this version may be that you need more generators for some 7-limit intervals but that's another story.

Let's take another example: 126/125. If I want this to vanish in a 2D system, I can either temper it out together with something else; or I can find a non-octave temperament which does the job; or I can spot that it's actually (6/5)^3 / (12/7)^1 and temper it out solely in terms of the two approximated intervals, 6/5 and 12/7, keep the 2/1 octave and still have it 2D. This means, of course, that triads like 3:5:7 would be possible in such a temperament but something like 3/1 wouldn't. Do we want to have a 3/1 there as well, yes? Okay, we'll add another generator and that gives us starling. Again, if you use generators of 3/1 and 5/1 instead, you'll need more of them for 7-limit intervals -- and I personally don't like this version of starling very much as it's clearly more complex than if one generator is a minor third.

And now the final revealing comes -- if we temper out both 225/224 and 126/125, we get 7-limit meantone and the aforementioned 3136/3125 vanishes as well. If we look at it in terms of the three commas, we find that we need 5 generators for 16/15, 3 generators for 6/5, and -4 generators for 5/4. This means that 7-limit meantone requires 5 times more generators than if we temper out 225/224 by itself, as long as we're only interested in 15/1 and 7/1, then it requires 3 times more generators than if we temper out 126/125 by itself, as long as we're only interested in 5/3 and 7/3, and it requires 2 times more generators than if we temper out 3136/3125 by itself, as long as we're only interested in 5/1 and 7/1. But it's perfectly okay as it can approximate 3/1, 5/1, and 7/1 at the same time. OTOH, if we do care about the non-prime factors without explicitly requiring the prime ones, we can temper out only one of the commas and sometimes even get a less complex temperament.

Petr

On Fri, Jun 24, 2011 at 9:24 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> However, if you want to find a 2D tuning tempering out
> > 703125/702464, then you have to approximate some other ratios.
> >
> > Why can't you still treat the generators as 3/1 and 5/1?
>
> Because you won't get all of the 7-limit intervals.

Oh, duh. OK. I didn't think this through enough and thought that using
2/1, 3/1, and 5/1 would be isomorphic to the ratios that Petr listed,
but I see I've screwed it up.

-Mike

On Sat, Jun 25, 2011 at 3:23 AM, petrparizek2000
<petrparizek2000@...> wrote:
>
> However, now let's suppose we have not three but two starting requirements: "Use 2/1 as the equivalence interval, find a 2D tuning which tempers out such and such a comma". If the vanishing interval contains three primes and one of them is two, then the procedure is obvious (for example, when tempering out 3136/3125, 2 generators approximate 5/4 and 5 generators approximate 7/4). However, what if it's something like 225/224, which contains 4 different primes (which would suggest to use 3 target approximants instead of 2, those being 3/1, 5/1, 7/1), and you still want a 2D temperament? Well, you have basically three options to choose from. #1: You can temper out two commas at a time, which makes it perfectly possible to start with 3 target approximated factors instead of 2. #2: You can make a non-octave temperament that would use a tempered 5/4 as the generator and a tempered 4/3 or 5/3 as the period (which is what I've done in my little "Non-octave marvel" study last year). #3: You can spot that 225/224 is (8/7)^1 / (16/15)^2; so you can happily use an octave as the period and a tempered minor second as the generator, two of which make a tempered 8/7. Note that in this third example, we're approximating 16/15 without actually including 3/2 or 5/4. That means that our target approximated odd factors now are 15/1 and 7/1 (I'm stripping the 2 out because that's our period). And this "option #3" is what I'm thinking of here

Oh ho ho ho ho! Very clever. So you recognize that, while 225/224
yields you a 3D temperament in the 2.3.5.7 limit, it yields you a 2D
temperament in the 2.7.15 limit. So your goal then is to find a way to
find useful 2D subgroups for 3D temperaments. We ran into the same
thing when looking at 91/90, 105/104, and 676/675, the latter of which
you've already explored.

The 2.5.9/7 subgroup would be another one for marvel temperament, I
think, right?

> Let's take another example: 126/125. If I want this to vanish in a 2D system, I can either temper it out together with something else; or I can find a non-octave temperament which does the job; or I can spot that it's actually (6/5)^3 / (12/7)^1 and temper it out solely in terms of the two approximated intervals, 6/5 and 12/7, keep the 2/1 octave and still have it 2D. This means, of course, that triads like 3:5:7 would be possible in such a temperament but something like 3/1 wouldn't. Do we want to have a 3/1 there as well, yes? Okay, we'll add another generator and that gives us starling. Again, if you use generators of 3/1 and 5/1 instead, you'll need more of them for 7-limit intervals -- and I personally don't like this version of starling very much as it's clearly more complex than if one generator is a minor third.

OK, so the least complexity basis that Gene's since developed takes
care of this then.

> And now the final revealing comes -- if we temper out both 225/224 and 126/125, we get 7-limit meantone and the aforementioned 3136/3125 vanishes as well. If we look at it in terms of the three commas, we find that we need 5 generators for 16/15, 3 generators for 6/5, and -4 generators for 5/4. This means that 7-limit meantone requires 5 times more generators than if we temper out 225/224 by itself, as long as we're only interested in 15/1 and 7/1, then it requires 3 times more generators than if we temper out 126/125 by itself, as long as we're only interested in 5/3 and 7/3, and it requires 2 times more generators than if we temper out 3136/3125 by itself, as long as we're only interested in 5/1 and 7/1. But it's perfectly okay as it can approximate 3/1, 5/1, and 7/1 at the same time. OTOH, if we do care about the non-prime factors without explicitly requiring the prime ones, we can temper out only one of the commas and sometimes even get a less complex temperament.

OK, I get it now. So the goal is to find subgroups for which a comma
gives you a rank-2 temperament of least complexity then. Not a bad
idea!

-Mike

On Fri, Jun 24, 2011 at 9:07 AM, Petr Parízek <petrparizek2000@...> wrote:
>
> Now let me tell you a sweet secret. Although I wasn't aware that it could
> work in this case, it's actually exactly the same procedure I've been using
> for finding 5-limit 2D temperaments. For example, the version with the
> lowest(abs) multipliers for amity is:
> [9 -13 5] = [39 -13 0] + [-30 0 5]
>
> Wow, I'm so happy that we've nailed it down in the end ... Now I suspect
> that there are some interesting 2D tunings hidden in these 7-limit factors.
> I'm only afraid it would take me whole days to get to them because finding
> the lowest multipliers by hand is terribly slow, or at least when I'm doing
> it -- but I've done it many times already so I'll do it a few times more --
> at least I'll refresh my math skills that way.and then, what comes next ...
> Music.

I'm bumping this post from several months ago because I've just
discovered that this is the greatest idea ever, and that Petr has the
answer to every question that's ever been asked about heirarchical
organizations of tonal pitch and all that. Well, OK, maybe not every
question, but this is a major piece of it. I take back any
disagreement I had with Petr about where tonality comes from. I've
been playing with 245/243-tempered 1/1-9/7-5/3 chords all night, and
1/1-6/5-36/25-7/4 chords as well, and it's tripping me out.

Mentally, we seem to build these little harmonic routes from 1/1 to
whatever note you hit, which could quite literally be said to be
higher-dimensional JI interval "functions." When two of these
functions merge - like a 7/4 up from 1/1, or also three 6/5's up - the
"quality" of the note somehow ends up being a mixture of the two
functions and merges into something new. For me, synesthetically, the
color that I see ends up being very literally a mixture of the two
colors of the original notes before merging. For example, in mavila,
5/2 is also 4/3 * 4/3 * 4/3 - the former is black, and the latter is
orange, and the result is like some kind of dark blood-orange color.
This is crazy and I have to keep pinching myself to make sure I'm
still alive, but it appears to be working.

For those who haven't ever experienced any degree of sound-color
synesthesia, more generally I'm talking about what appears to be
"chroma" formation now. I have no idea how you all without AP perceive
music, but I note that "chroma" and "function" seem to be related, so
I substituted function for chroma in the above paragraph.

I've spent 4 years on this list now, and everything is now starting to
make sense. I spoke to Petr before about developing a "proto-harmonic"
system, and Petr said he preferred to think in tempered generator
systems, but now I'm happy because I see how the two are related. As
always, I encourage you all to weigh in with your own experiences as
well.

I have 2 questions for Petr:
1) How did you figure this out?
2) Do you feel like codimension-1 temperaments will be easier to
understand than codimension-2? i.e. do you think that a rank-2 7-limit
subgroup temperament, eliminating only 1 comma, will be easier to
figure out than a rank-2 full 7-limit temperament eliminating 2
commas?

I'm also curious if Gene, who I remember was uploading a lot of these
illusory chords on the wiki a while ago, was doing it for this reason
as well.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I'm also curious if Gene, who I remember was uploading a lot of these
> illusory chords on the wiki a while ago, was doing it for this reason
> as well.

No, it was because I think they are very useful kinds of chords, adding fluidity to harmonic relationships. But much of the time I can't understand your postings, and this is one of those times, so I don't know what else to say.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > I'm also curious if Gene, who I remember was uploading a lot of these
> > illusory chords on the wiki a while ago, was doing it for this reason
> > as well.
>
> No, it was because I think they are very useful kinds of chords, adding fluidity to harmonic relationships. But much of the time I can't understand your postings, and this is one of those times, so I don't know what else to say.

I think this is pretty funny.

Keenan

On Sep 7, 2011, at 11:20 AM, "genewardsmith" <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I'm also curious if Gene, who I remember was uploading a lot of these
> illusory chords on the wiki a while ago, was doing it for this reason
> as well.

No, it was because I think they are very useful kinds of chords, adding
fluidity to harmonic relationships. But much of the time I can't understand
your postings, and this is one of those times, so I don't know what else to
say.

I'm saying that I find that decently accurate rank-2 codimension-1 subgroup
temperaments that are low in complexity really trip me out. I find that
it's easy to internalize the tempered comma, much like we've internalized
81/80 vanishing in meantone, and it seems to be the key to a really strong
novel experience, at least for me.

For example, in meantone, we're intuitively aware that 5/4 = 9/8 * 9/8, and
it colors the sound of the 5/4, turning the whole thing into the more
complex gestalt of the meantone "major third." I find that the same thing
happens with 5/3 being 9/7 * 9/7 when 245/243 vanishes, which creates these
whooshing 1/1-9/7-5/3 chords that I think sound really xenharmonic. The same
applies to the 1/1-9/7-13/8 chords in 17-equal, which sound gray and purple
at the same time, like that really nice outfit my friend Mel was wearing at
UM 3 years ago. Or 875/864 tempered 1/1-6/5-36/25-7/4 chords which are
diminished, yet partly otonal and "rooted" at the same time, profoundly
transforming the experience of a diminished chord into something new and
exciting and transcendent. It's a rather pleasant experience to be having.

Most importantly, to me, it seems to suggest something like proto-tonality,
where I can start to hear more complex notes in sensi[8] as being compound
intervals reachable by simpler consonances like 9/7 and 5/3, kind of how I
hear the meantone major sevenths as being reachable by fifth+third. (This
assumes your ear is trained enough to hear 9/7 as not just a sharp 5/4). So
I feel like this is only one step away from a full-blown tonal heirarchy,
and now I can relate a lot more to Petr's work from a month ago.

So I hope you might find this information useful enough to mathematically
generalize, which is why I mentioned you.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I'm saying that I find that decently accurate rank-2 codimension-1 subgroup
> temperaments that are low in complexity really trip me out. I find that
> it's easy to internalize the tempered comma, much like we've internalized
> 81/80 vanishing in meantone, and it seems to be the key to a really strong
> novel experience, at least for me.
>
> For example, in meantone, we're intuitively aware that 5/4 = 9/8 * 9/8, and
> it colors the sound of the 5/4, turning the whole thing into the more
> complex gestalt of the meantone "major third." I find that the same thing
> happens with 5/3 being 9/7 * 9/7 when 245/243 vanishes, which creates these
> whooshing 1/1-9/7-5/3 chords that I think sound really xenharmonic. The same
> applies to the 1/1-9/7-13/8 chords in 17-equal, which sound gray and purple
> at the same time, like that really nice outfit my friend Mel was wearing at
> UM 3 years ago. Or 875/864 tempered 1/1-6/5-36/25-7/4 chords which are
> diminished, yet partly otonal and "rooted" at the same time, profoundly
> transforming the experience of a diminished chord into something new and
> exciting and transcendent. It's a rather pleasant experience to be having.
>
> Most importantly, to me, it seems to suggest something like proto-tonality,
> where I can start to hear more complex notes in sensi[8] as being compound
> intervals reachable by simpler consonances like 9/7 and 5/3, kind of how I
> hear the meantone major sevenths as being reachable by fifth+third. (This
> assumes your ear is trained enough to hear 9/7 as not just a sharp 5/4). So
> I feel like this is only one step away from a full-blown tonal heirarchy,
> and now I can relate a lot more to Petr's work from a month ago.
>
> So I hope you might find this information useful enough to mathematically
> generalize, which is why I mentioned you.

To me, this feeling you describe has always been a major part of "the point of temperament". Are you saying this works better with codimension-1 temperaments?

What do you mean when you say sensi is codimension-1? Do you mean you're treating it as a 2.9/7.5/3 subgroup temperament?

Keenan

On Wed, Sep 7, 2011 at 2:33 PM, Keenan Pepper <keenanpepper@...> wrote:
>
> To me, this feeling you describe has always been a major part of "the point of temperament". Are you saying this works better with codimension-1 temperaments?

I'm not sure if it really works better, but codimension-1 is a good
starting point for me to understand what's going on, because it's the
simplest type of temperament possible. Something like a
245/243-tempered 1/1-9/7-5/3 chord is a neat little harmonic unit,
kind of like C-E-G-A-D in meantone. Paul has called them "necessarily
tempered chords" and Carl has called them "magic chords." An 11-limit
temperament that's codimension-3 will also be obviously usable, but
there's just more harmonic relationships to figure out, so I'm finding
it useful to step back and start with tempering out single commas
first.

It's a new paradigm for me because I've been trying to use
temperaments to approximate higher-limit otonalities, but something
like 1/1-9/7-5/3 isn't an otonality. But it still works, and in a much
trippier way.

I'm starting to realize is that this doesn't just apply to chords, but
to learning how to relate more temperamentally-complex intervals back
to the root as being "compound" intervals - kind of how although a
major seventh is really dissonant, once you learn it's a major third +
a perfect fifth it's not so bad. But I'm still not sure how this all
works, so I'll leave it alone for now.

> What do you mean when you say sensi is codimension-1? Do you mean you're treating it as a 2.9/7.5/3 subgroup temperament?

Yes, I wasn't talking about sensi, just the 245/243 linear temperament
that you get if you use Petr's formula, which is 2.9/7.5/3.

-Mike

On Wed, Sep 7, 2011 at 2:22 PM, Keenan Pepper <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > I'm also curious if Gene, who I remember was uploading a lot of these
> > > illusory chords on the wiki a while ago, was doing it for this reason
> > > as well.
> >
> > No, it was because I think they are very useful kinds of chords, adding fluidity to harmonic relationships. But much of the time I can't understand your postings, and this is one of those times, so I don't know what else to say.
>
> I think this is pretty funny.

LOL, I'm working alone, in a vacuum. Space is infinite and cold and
lonely, it seems.

-Mike 9000

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Paul has called them "necessarily
> tempered chords" and Carl has called them "magic chords." An 11-limit
> temperament that's codimension-3 will also be obviously usable, but
> there's just more harmonic relationships to figure out, so I'm finding
> it useful to step back and start with tempering out single commas
> first.

I suppose you know about this page:

http://xenharmonic.wikispaces.com/Dyadic+chord

On Sep 7, 2011, at 7:28 PM, genewardsmith <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Paul has called them "necessarily
> tempered chords" and Carl has called them "magic chords." An 11-limit
> temperament that's codimension-3 will also be obviously usable, but
> there's just more harmonic relationships to figure out, so I'm finding
> it useful to step back and start with tempering out single commas
> first.

I suppose you know about this page:

http://xenharmonic.wikispaces.com/Dyadic+chord

Yep and I'll spend some time on it tonight, but if you read my original
message, I was curious if you were drawn to these chords for the same reason
I am, and I'm still curious about that.

-Mike

On Sep 7, 2011, at 7:28 PM, genewardsmith <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Paul has called them "necessarily
> tempered chords" and Carl has called them "magic chords." An 11-limit
> temperament that's codimension-3 will also be obviously usable, but
> there's just more harmonic relationships to figure out, so I'm finding
> it useful to step back and start with tempering out single commas
> first.

I suppose you know about this page:

http://xenharmonic.wikispaces.com/Dyadic+chord

Some of these are unbelievably trippy. I still don't know if anyone else is
hearing this all like I described, but I'm hearing random dyads converging
on one another and the colors fusing and it's beautiful.

If anyone else hears these chords in a similar way, I also suggest 96/95
might be theoretically important. 57/50 as a close runner pmn

--- In tuning@yahoogroups.com, Mike 9000 wrote:
> LOL, I'm working alone, in a vacuum. Space is infinite and cold and
> lonely, it seems.

Mike, with your 1/1 - 9/7 - 5/3 triad I guess you are making the lower and upper intervals equal? and using this generator to make 8- or 13- note MOS? If so, you could try (25-sqrt(5))/62 *1200cents (slightly more than 440 cents) as the generator - this gives L/s = phi ... what do you think?

Also, what do you think of that triad widened to lower=upper=457.9cents? This generator gives 8- or 11-note MOS.

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike 9000 wrote:
> > LOL, I'm working alone, in a vacuum. Space is infinite and cold and
> > lonely, it seems.
>
> Mike, with your 1/1 - 9/7 - 5/3 triad I guess you are making the lower and upper intervals equal? and using this generator to make 8- or 13- note MOS? If so, you could try (25-sqrt(5))/62 *1200cents (slightly more than 440 cents) as the generator - this gives L/s = phi ... what do you think?
>
> Also, what do you think of that triad widened to lower=upper=457.9cents? This generator gives 8- or 11-note MOS.
>
> Steve M.
>
Ah, no, swap the 13- and the 11- in the above...

On Fri, Sep 9, 2011 at 1:04 PM, martinsj013 <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike 9000 wrote:
> > LOL, I'm working alone, in a vacuum. Space is infinite and cold and
> > lonely, it seems.
>
> Mike, with your 1/1 - 9/7 - 5/3 triad I guess you are making the lower and upper intervals equal? and using this generator to make 8- or 13- note MOS? If so, you could try (25-sqrt(5))/62 *1200cents (slightly more than 440 cents) as the generator - this gives L/s = phi ... what do you think?

Yep. This chord has 245/243 vanish, and is a canonical chord in sensi
temperament. As for the tuning, I think it's great! Stacked 9/7's
sound really dissonant at first, almost like augmented, but once you
play them a lot they start to change. They sound like they produce
this "whoooshing" sound when you move them up and down the scale
diatonically. It's good stuff.

The sLssLsLs mode is great because it's also "tonal" - has a tritone
that resolves outward to 1/1-9/7-5/3. Play it, it really works. Now we
can do tonal sensi temperament too. Once we figure out the modal stuff
it's all over.

> Also, what do you think of that triad widened to lower=upper=457.9cents? This generator gives 8- or 11-note MOS.

It's crazy you're talking about this, because I'm just playing in
13-equal now. This forms 5L3s scales. People hate them because the
fifths are so sharp, but I -love- them. Like -LOVE- them. They're a
whole new thing. Once you get used to the sharp fifths, it's one of
the most xenharmonic sounds possible.

Try it in 13-equal, just play the 738 cent circle of fifths. Three of
them puts you at 9/5, and four of them puts you at 11/8
(octave-equivalent), five gets you to 17/16, six gets you to 13/8.
It's just insane. The sharp fifths lead you into all of these
ridiculously xenharmonic intervals, but nobody uses them because we're
all afraid of 3/2 being that out of tune.

Maybe we just need to find a decent subgroup mapping for around 740
cents. This is a seriously awesome temperament. The version mapping
the generator to 3/2 I'm calling it "coral temperament," just trying
to work out a comma basis for it now.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Sep 9, 2011 at 1:04 PM, martinsj013 <martinsj@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike 9000 wrote:
> > > LOL, I'm working alone, in a vacuum. Space is infinite and cold and
> > > lonely, it seems.
> >
> > Mike, with your 1/1 - 9/7 - 5/3 triad I guess you are making the lower and upper intervals equal? and using this generator to make 8- or 13- note MOS? If so, you could try (25-sqrt(5))/62 *1200cents (slightly more than 440 cents) as the generator - this gives L/s = phi ... what do you think?
>
> Yep. This chord has 245/243 vanish, and is a canonical chord in sensi
> temperament.

http://xenharmonic.wikispaces.com/sensamagic+triad

Mike wrote:

> 1) How did you figure this out?

What in particular did you mean?

> 2) Do you feel like codimension-1 temperaments will be easier to
> understand than codimension-2? i.e. do you think that a rank-2 7-limit
> subgroup temperament, eliminating only 1 comma, will be easier to
> figure out than a rank-2 full 7-limit temperament eliminating 2
> commas?

I'm afraid I'm unable to answer your question because I've never tried to think that way when applying the idea. If I want to find a 2D tuning which tempers out, let's say, 10976/10935, then ... Well, then ... Ehh, I'll leave it for you to ponder.

Petr

On Sun, Sep 11, 2011 at 6:49 PM, petrparizek2000
<petrparizek2000@...> wrote:
>
> Mike wrote:
>
> > 1) How did you figure this out?
>
> What in particular did you mean?

We had a discussion a while ago about understanding "proto-harmonic
systems" and then understanding the "tempered harmonic system." You
said you used to think in terms of a "proto-harmonic system" and then
you shifted to thinking purely in terms of temperament, focusing on
the generator chain, because you became enlightened and realized that
this can help you learn how harmony works. How did you come to realize
that?

BTW, I think that there are still some advantages of understanding
"proto-harmony," so long as you understand how "proto-harmony" morphs
into tempered harmony. Proto-harmony is really just 5-limit JI,
anyway, so it's just another tuning system. Once you understand how
5-limit JI is related and differs from its tempered derivatives, you
can start to understand commonalities across tuning systems, which I
think is great.

> > 2) Do you feel like codimension-1 temperaments will be easier to
> > understand than codimension-2? i.e. do you think that a rank-2 7-limit
> > subgroup temperament, eliminating only 1 comma, will be easier to
> > figure out than a rank-2 full 7-limit temperament eliminating 2
> > commas?
>
> I'm afraid I'm unable to answer your question because I've never tried to think that way when applying the idea. If I want to find a 2D tuning which tempers out, let's say, 10976/10935, then ... Well, then ... Ehh, I'll leave it for you to ponder.

By "codimension" I meant the number of commas being tempered out. For
meantone, porcupine, negri, etc, there's only one fundamental tempered
relationship you have to learn, as well as all of its implications.
But in 11-limit Orwell, for example, there's four fundamental tempered
relationships you have to learn, and all of the implications just run
right on top of one another. Do you think it matters? I thought so at
first, but now I'm not sure it does.

-Mike

Mike wrote:

> We had a discussion a while ago about understanding "proto-harmonic
> systems" and then understanding the "tempered harmonic system." You
> said you used to think in terms of a "proto-harmonic system" and then
> you shifted to thinking purely in terms of temperament, focusing on
> the generator chain, because you became enlightened and realized that
> this can help you learn how harmony works. How did you come to realize
> that?

I see. That's quite a long story but I'll try to make it shorter.

The first new "recognition" occurred to me around February 2006 when I managed to get some sort of "systematic understanding" of the non-octave temperament where 245/243 vanishes. Therefore, I was finally able to grasp why the so-called "BP diatonic" was said to have 9 steps to a 3/1 and why a "BP chromatic" was said to have 13 steps and so on. I gradually learned to understand the concept of non-octave non-equal temperaments when I later examined the one whose period was 4/1 and whose generator was the 9th root of 32/5 -- or, in the weirdest case, the one whose period was 9/5 and whose generator was an almost 9/7 so that 2430/2401 went tempered out. (Unfortunately, I never had a chance to try them out in real-time.) For finding all of these, I was using the method described in my previous message.

Then, a major shift in my thinking came in June the same year, when I essentially rediscovered semisixths and hanson almost by accident. That made me think: "Oh my goodness, now all those 3/2s and 5/4s have totaly different musical meanings than what we're used to, this is just twisted." Maybe I wouldn't arrive at that conclusion if I hadn't been playing in real-time that moment, who knows? Anyway, the way I first arrived at semisixths also influenced my understanding a lot -- as I was interested in BP those days, I first made a scale using a period of 3/1 and I realized that I could actually get octaves in there if the generator were slightly wider than my preferred choice. Then I thought: "What about reducing this to 2/1 rather than 3/1?" And there I had it.

Later, I found that I could actually get a temperament just by tempering out a particular comma -- like, for example, starting with a monzo like "9 -13 5" and turning it into a pair of monzos like "39 -13 0" and "-30 0 5", which gives me some rough notion about a possible generator and also tells me something about the mapping. That changed my understanding of the entire topic of 2D temperaments an awful lot because I finally started to be aware of not only the melodic properties of a particular tuning but also the harmonic ones.

Finally, when I got involved into the matter of comma pumps to an unexpectedly large extent, all of my previous conclusions about completely new harmonic systems seemed to me to be supported once again -- even in situations when I first thought that generator distances wouldn't play such a big role; and in the end they did. It's not easy to say in words, one just has to play around with it and ake a few pieces with it.

> By "codimension" I meant the number of commas being tempered out. For
> meantone, porcupine, negri, etc, there's only one fundamental tempered
> relationship you have to learn, as well as all of its implications.
> But in 11-limit Orwell, for example, there's four fundamental tempered
> relationships you have to learn, and all of the implications just run
> right on top of one another. Do you think it matters? I thought so at
> first, but now I'm not sure it does.

You can never say that it *is* or that it *isn't* important; it depends on whether you care or not. For example, is quarter-comma meantone a 5-limit temperament? Or is it a 7-limit temperament? Or is it an 11-limit temperament, for that matter? Or is it a 3136/3125 temperament with an added approximation of 3/2?

BTW: Why should there be 4 basic relationships for an 11-limit temperament?

Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> [tempered 1/1 - 9/7 - 5/3 with lower and upper intervals equal]
> [8- or 11- note MOS]
> This chord has 245/243 vanish, and is a canonical chord in sensi
> temperament.

Mike,
If you're interested, I've uploaded a contour map of 3HE near this triad, see:
/tuning/files/g4c5c1M.png

Some rational triads are labelled, also some other points needing explanation (my lazy programming means that only integers are accepted in the labels right now):

3:0:5 - actually 3:sqrt(15):5 (lower=upper=sqrt(5/3))
0:5:13 - actually lower=upper=2^(5/13)
0:4:13 - actually lower=upper=2^(4/13)
0:13:13 - actually lower=upper=457.9cents, from my post.

My 457.9cents (ratio (sqrt(13)-1)/2) was chosen because it is on a "channel" of comparatively low 3HE.

> The sLssLsLs mode is great because it's also "tonal" - has a tritone
> that resolves outward to 1/1-9/7-5/3. Play it, it really works.
Not had time yet, but I will.

Steve.

I wrote:

> Finally, when I got involved into the matter of comma pumps to an > unexpectedly
> large extent, all of my previous conclusions about completely new harmonic
> systems seemed to me to be supported once again -- even in situations when > I
> first thought that generator distances wouldn't play such a big role; and > in the
> end they did. It's not easy to say in words, one just has to play around > with it

Or let's put int another way ... If I asked you to find a chord progression where the first and the last chord differ by 78732/78125, what would you do then?

Petr