Miracle theory

The "miracle" generator splits the difference between 15/14 and
16/15, and hence is associated to the scales with the jumping jack
225/224 in the kernel. If we seek other generators with miracle
properties, and obvious place to began is with other jumping jacks.
If we look instead at 81/80, we get the meantone miracle, which is
well-known. If we look higher, we find 2401/2400 = (49/48)(49/50),
3025/3024 = (55/54)(55/56), and so forth. We therefore might expect a
miracle generator somewhere in the interval between 50/49 and 49/48,
which is to say somewhere between 35 and 35.7 cents.

Checking ets with 2401/2400 in the kernel and with good properties,
we find that 18/612 = 35.3 cents and 13/441 = 35.37 cents, which
gives us a pretty good notion of where the next miracle is hiding. On
the high side we have 8/270 = 35.555 cents, and on the low side 5/171
= 10/342 = 35.08 cents.

The meantone miracle is really better viewed as a flat fifth miracle;
we have 9/8 = (3/2)^2 1/2, so that by extending the circle of fifths
and reducing by octaves we obtain the true miracle of the meantone.
It seems to me it would make sense to do likewise with these other
miracles; we have 16/15 = (4/3)^2 3/5, so that a circle of miracle
fourths in a scale whose interval of repetition is the major sixth
would seem to be the way to take full advantage of the miracle
generator. In the same way, for the 35-cent miracle, we have
49/48 = (7/4)^2 1/3, so a circle of 7/4's with an interval of
repetition of 3 would make for an interesting collection of scales.

--- In tuning-math@y..., genewardsmith@j... wrote:
> The "miracle" generator splits the difference between 15/14 and
> 16/15, and hence is associated to the scales with the jumping jack
> 225/224 in the kernel. If we seek other generators with miracle
> properties, and obvious place to began is with other jumping jacks.
> If we look instead at 81/80, we get the meantone miracle, which is
> well-known.

The meantone miracle?

If we look higher, we find 2401/2400 = (49/48)(49/50),
> 3025/3024 = (55/54)(55/56), and so forth. We therefore might expect
a
> miracle generator somewhere in the interval between 50/49 and
49/48,
> which is to say somewhere between 35 and 35.7 cents.

What do you mean by miracle generator here? Miracle generator for us
has meant specifically 116.7 cents . . .
>
> Checking ets with 2401/2400 in the kernel and with good properties,
> we find that 18/612 = 35.3 cents and 13/441 = 35.37 cents, which
> gives us a pretty good notion of where the next miracle is hiding.

You need to define your conception of "miracle" for us.

On
> the high side we have 8/270 = 35.555 cents, and on the low side
5/171
> = 10/342 = 35.08 cents.
>
> The meantone miracle is really better viewed as a flat fifth
miracle;
> we have 9/8 = (3/2)^2 1/2, so that by extending the circle of
fifths
> and reducing by octaves we obtain the true miracle of the meantone.
> It seems to me it would make sense to do likewise with these other
> miracles; we have 16/15 = (4/3)^2 3/5, so that a circle of miracle
> fourths in a scale whose interval of repetition is the major sixth
> would seem to be the way to take full advantage of the miracle
> generator.

Would the octave no longer be an interval of repetition in your view?
Can you clarify at all? What do you meant by "take full advantage"
that may be different from our previous focus on number of consonant
chords?

I found this comment from Paul in my inbox:

> What do you mean by miracle generator here? Miracle generator for us
> has meant specifically 116.7 cents . . .

I don't know about the rest of you, but I take anything between 3/31
and 4/41 octaves as a Miracle generator. That's from 116.1 to 117.1
cents. 16:15 is 111.7 cents and 15:14 is 119.4 cents.

Graham

--- In tuning-math@y..., graham@m... wrote:

> I don't know about the rest of you, but I take anything between
3/31
> and 4/41 octaves as a Miracle generator. That's from 116.1 to
117.1
> cents. 16:15 is 111.7 cents and 15:14 is 119.4 cents.

Splitting the difference gives us sqrt(8/7) = 115.6, a little shy of
being a miracle generator; however on the tuning group I pointed out
that m = (12/5)^(1/13) = 116.6 cents is what gives us perfect octaves
if we iterate a miracle fourth inside of 5/3, and it is something in
the vicinity of m I would regard as a miracle generator. The
corresponding miracle fourth is f = sqrt(5/3 m) =
(2 (5/3)^6)^(1/13) = 500.5 cents, and it is this that I regard as
more fundamental, the miracle generator itself being secondary.

Incidentally, the difference between sqrt(8/7) and m, which is
(2^35 3^4 5^(-2) 7^13)^(1/26), is 1.0007446 cents, which must be the
justification for using cents. :)

Gene wrote:

> Splitting the difference gives us sqrt(8/7) = 115.6, a little shy of
> being a miracle generator; however on the tuning group I pointed out
> that m = (12/5)^(1/13) = 116.6 cents is what gives us perfect octaves
> if we iterate a miracle fourth inside of 5/3, and it is something in
> the vicinity of m I would regard as a miracle generator. The
> corresponding miracle fourth is f = sqrt(5/3 m) =
> (2 (5/3)^6)^(1/13) = 500.5 cents, and it is this that I regard as
> more fundamental, the miracle generator itself being secondary.

That generator is also what gives you a perfect 5:3 when you iterate it
within a 2:1. And it's well within the vicinity *I* would regard as a
miracle generator. But I don't see what's special about it, or why your
definition is secondary to the usual Secors-within-an-octave one. For
meantone pair, the transformation would have to be

(tone) = (2 -1)(fifth)
(oct ) (0 1)(oct )

which has a determinant of 2, because you can't get a fifth by adding and
subtracting tones and octaves. The miracle equivalent you give

(fourth) = ( -6 1)(Secor)
(sixth ) (-13 2)(oct )

is unitary, and so at least works.

(Secor) = ( 2 -1)(fourth)
(oct ) (13 -6)(sixth )

But why more fundamental?

Graham

--- In tuning-math@y..., graham@m... wrote:

> That generator is also what gives you a perfect 5:3 when you
iterate it
> within a 2:1. And it's well within the vicinity *I* would regard
as a
> miracle generator. But I don't see what's special about it, or why
your
> definition is secondary to the usual Secors-within-an-octave one.

I explained this on
/tuning/topicId_28290.html#28290, or at least so I
hope. I think the Secors version is secondary for the same reason
that iterating a meantone is a less important scale generating
process than iterating a meantone tempered fifth--the scales you get
from the flat fifth contain all the intervals you normally want, not
just some of them. I discussed this analogy in the article
/tuning/topicId_28335.html#28335,
and the next miracle I mentioned in
/tuning/topicId_28465.html#28465.

As for the exact value, it is no more magical than the exact
1/4-comma meantone is magical. However, just as that meantone gives
you exactly a 5 when interating within 2s, the above value is what
you need to get an exact 2 when iterating within 5/3.

> (fourth) = ( -6 1)(Secor)
> (sixth ) (-13 2)(oct )
>
> is unitary, and so at least works.

Unimodular is the word you want, unitary means that the inverse
matrix is the complex conjugate of the transpose of the matrix.

Gene wrote:

> I explained this on
> /tuning/topicId_28290.html#28290, or at least so I
> hope.

That's "More on Miracles". It explains the arithmetic, but not the
reasons for it.

> I think the Secors version is secondary for the same reason
> that iterating a meantone is a less important scale generating
> process than iterating a meantone tempered fifth--the scales you get
> from the flat fifth contain all the intervals you normally want, not
> just some of them.

But in this case *both* methods give all the intervals you want!

> I discussed this analogy in the article
> /tuning/topicId_28335.html#28335,

That's "A meantone analogue of Blackjack". It shows a scale generated by
the whole tone.

> and the next miracle I mentioned in
> /tuning/topicId_28465.html#28465.

This doesn't make much sense to me.

> As for the exact value, it is no more magical than the exact
> 1/4-comma meantone is magical. However, just as that meantone gives
> you exactly a 5 when interating within 2s, the above value is what
> you need to get an exact 2 when iterating within 5/3.

But the value for quarter comma meantone is the minimax for the 5-limit.
There doesn't seem to be an analogy here.

> Unimodular is the word you want, unitary means that the inverse
> matrix is the complex conjugate of the transpose of the matrix.

Oops, thanks.

Graham

genewardsmith@juno.com () wrote:

> The "miracle" generator splits the difference between 15/14 and
> 16/15, and hence is associated to the scales with the jumping jack
> 225/224 in the kernel. If we seek other generators with miracle
> properties, and obvious place to began is with other jumping jacks.
> If we look instead at 81/80, we get the meantone miracle, which is
> well-known. If we look higher, we find 2401/2400 = (49/48)(49/50),
> 3025/3024 = (55/54)(55/56), and so forth. We therefore might expect a
> miracle generator somewhere in the interval between 50/49 and 49/48,
> which is to say somewhere between 35 and 35.7 cents.

225:224 is also in the kernel for characteristic meantone, schismic and
"magic" scales. There's nothing special about associating it with
miracles. 225:224, 2401:2400 and 3025:3024, however, completely define
miracle temperament. Add 81:80, and you have 31-equal (actually a 62 note
periodicity block). So is 31-equal the grand unified jumping jack
temperament?

> Checking ets with 2401/2400 in the kernel and with good properties,
> we find that 18/612 = 35.3 cents and 13/441 = 35.37 cents, which
> gives us a pretty good notion of where the next miracle is hiding. On
> the high side we have 8/270 = 35.555 cents, and on the low side 5/171
> = 10/342 = 35.08 cents.

I don't get this. The temperament consistent with 441- and 612-equal
divides the octave into 9 equal parts, with a generator mapping of [-2 -3
-2]. The generator is around 49 cents, 18/441 or 25/612 octaves. It
covers the 9-limit with 37 notes, with 0.2 cents accuracy.

> The meantone miracle is really better viewed as a flat fifth miracle;
> we have 9/8 = (3/2)^2 1/2, so that by extending the circle of fifths
> and reducing by octaves we obtain the true miracle of the meantone.
> It seems to me it would make sense to do likewise with these other
> miracles; we have 16/15 = (4/3)^2 3/5, so that a circle of miracle
> fourths in a scale whose interval of repetition is the major sixth
> would seem to be the way to take full advantage of the miracle
> generator. In the same way, for the 35-cent miracle, we have
> 49/48 = (7/4)^2 1/3, so a circle of 7/4's with an interval of
> repetition of 3 would make for an interesting collection of scales.

I don't see why these are better views, other than for meantone. As I
make the last case, the interval of repetition is 3, and a 7:4 is two
generators. 5:3 and 7:5 are both a single generator.

Graham

--- In tuning-math@y..., graham@m... wrote:

> > I think the Secors version is secondary for the same reason
> > that iterating a meantone is a less important scale generating
> > process than iterating a meantone tempered fifth--the scales you
get
> > from the flat fifth contain all the intervals you normally want,
not
> > just some of them.

> But in this case *both* methods give all the intervals you want!

The miracle generator certainly does better than the meantone, which
may be why it is a miracle, but I am not convinced. The JI diatonic
scale requires 25 miracle steps from 5/3 at -13 to 9/8 at 12, whereas
the compass is 14 for the fourth-in-5/3 version. Paul was actually
worried that Jacky's Blackjacky sounded too diatonic, an issue which
would not even arise in the other system.

> > As for the exact value, it is no more magical than the exact
> > 1/4-comma meantone is magical. However, just as that meantone
gives
> > you exactly a 5 when interating within 2s, the above value is
what
> > you need to get an exact 2 when iterating within 5/3.

> But the value for quarter comma meantone is the minimax for the 5-
limit.
> There doesn't seem to be an analogy here.

I don't know what you mean by "the" minimax, nor why you could not do
a similar calculation here.

--- In tuning-math@y..., graham@m... wrote:

> 225:224 is also in the kernel for characteristic meantone, schismic
and
> "magic" scales. There's nothing special about associating it with
> miracles.

I dunno--it's associated with miracles in something like the way
81/80 is associated to meantone, and that is a completely defining
way. But it seems you want more:

225:224, 2401:2400 and 3025:3024, however, completely define
> miracle temperament.

This means that h10 and h31 also completely define it; and we have
h41 = h10 + h31 and h72 = h10 + 2 h31 in this miracle kernel.

Add 81:80, and you have 31-equal (actually a 62 note
> periodicity block). So is 31-equal the grand unified jumping jack
> temperament?

It would be better if it came out 31 and not 62, but you do get 2 h31
out of the first four jumping jacks. I've been planning to write a
program to find more of these, and it will be interesting to see what
they lead to.

> I don't get this. The temperament consistent with 441- and 612-
equal
> divides the octave into 9 equal parts, with a generator mapping of
[-2 -3
> -2]. The generator is around 49 cents, 18/441 or 25/612 octaves.
It
> covers the 9-limit with 37 notes, with 0.2 cents accuracy.

Sounds interesting, I will check this out.

> I don't see why these are better views, other than for meantone.
As I
> make the last case, the interval of repetition is 3, and a 7:4 is
two
> generators. 5:3 and 7:5 are both a single generator.

I'm not sure what you mean by this--7/4 and 3 are two generators,
just as 3/2 and 2 are two generators or 5/3 and 4/3 are two
generators. Wherein lies the difference?

In-Reply-To: <9omkht+amc8@eGroups.com>
Gene wrote:

> > 225:224 is also in the kernel for characteristic meantone, schismic
> and
> > "magic" scales. There's nothing special about associating it with
> > miracles.
>
> I dunno--it's associated with miracles in something like the way
> 81/80 is associated to meantone, and that is a completely defining
> way. But it seems you want more:

81:80 completely defines meantone because it converts the three
dimensional 5-limit into a two-dimensional linear temperament. (Or 2-D
into 1-D with octave equivalence.) One dimension is being lost, so you
need one commatic unison vector. 225:224 is a 7-prime limit interval, and
can't of itself define a linear temperament.

> 225:224, 2401:2400 and 3025:3024, however, completely define
> > miracle temperament.
>
> This means that h10 and h31 also completely define it; and we have
> h41 = h10 + h31 and h72 = h10 + 2 h31 in this miracle kernel.

Yes, that's right.

> > I don't see why these are better views, other than for meantone.
> As I
> > make the last case, the interval of repetition is 3, and a 7:4 is
> two
> > generators. 5:3 and 7:5 are both a single generator.
>
> I'm not sure what you mean by this--7/4 and 3 are two generators,
> just as 3/2 and 2 are two generators or 5/3 and 4/3 are two
> generators. Wherein lies the difference?

No, 7:4 and 3 are two intervals, but they aren't generators of this scale.
The 3 is wrong anyway. I meant to say that the interval of equivalence
is 1/9 of an octave, whereas you seemed to say 1/3. 5:3 and 7:5 are both
equivalent to each other and to the generator in this system. 7:4 is not
a generator -- it's represented by two generator steps, the same as 9:8 in
meantone. The system you describe is different to the one I get by
following your instructions, so I don't understand where it comes from.

Graham

In-Reply-To: <9olut9+7mhg@eGroups.com>
Gene wrote:

> The miracle generator certainly does better than the meantone, which
> may be why it is a miracle, but I am not convinced. The JI diatonic
> scale requires 25 miracle steps from 5/3 at -13 to 9/8 at 12, whereas
> the compass is 14 for the fourth-in-5/3 version. Paul was actually
> worried that Jacky's Blackjacky sounded too diatonic, an issue which
> would not even arise in the other system.

Why is the JI diatonic being taken as the standard? Even so, your
calculation only seems to cover one octave. Merely to represent two
octaves with the 5:3 equivalence, you need 26 fourths. With octave
equivalence, any number of octaves can be represented with the same number
of miracle steps. With the 5:3 equivalence, an 11:9 is 43 fourths, as
compared to 3 Secors. So a neutral third scale will come out a lot worse
in your 5:3 system.

I thought Jacky's piece sounded too diatonic because the retuning wasn't
working.

> > But the value for quarter comma meantone is the minimax for the 5-
> limit.
> > There doesn't seem to be an analogy here.
>
> I don't know what you mean by "the" minimax, nor why you could not do
> a similar calculation here.

The 5-limit minimax is the tuning where the largest 5-limit error
(relative to JI) is as small as possible. The relevant limit for miracle
is 11, and indeed Dave Keenan did this calculation a long time ago. The
result is a 116.7 cent generator giving a worst error of 3.3 cents.

Graham

--- In tuning-math@y..., genewardsmith@j... wrote:

> Paul was actually
> worried that Jacky's Blackjacky sounded too diatonic, an issue
which
> would not even arise in the other system.

Huh? What do you mean? It's pretty clear that the initial pitch bend
messages did not take effect, leading to the beginning of the piece
being in 12-tET. What does this have to do with anything?

--- In tuning-math@y..., graham@m... wrote:

> No, 7:4 and 3 are two intervals, but they aren't generators of this
scale.

They are the generators of a system of scales I mentioned, so I need
a context for "this scale"--which scale?

In-Reply-To: <9oo0rt+s4tp@eGroups.com>
Gene wrote:

> > No, 7:4 and 3 are two intervals, but they aren't generators of this
> scale.
>
> They are the generators of a system of scales I mentioned, so I need
> a context for "this scale"--which scale?

>>> temper.Temperament(441,612,temper.primes[:-2])

43/117, 49.0 cent generator

basis:
(0.1111111111111111, 0.040838783185694151)

mapping by period and generator:
[(9, 0), (15, -2), (22, -3), (26, -2)]

mapping by steps:
[(612, 441), (970, 699), (1421, 1024), (1718, 1238)]

unison vectors:
[[-5, -1, -2, 4], [-1, -7, 4, 1]]

highest interval width: 3
complexity measure: 27 (45 for smallest MOS)
highest error: 0.000170 (0.204 cents)
unique

So which scale do you mean?

Graham

In-Reply-To: <9oo2ic+tpk6@eGroups.com>
Gene wrote:

> > Why is the JI diatonic being taken as the standard?
>
> Mostly because of the Blackjacky discussion. What would you suggest
> as a standard?

The 11-limit tonality diamond. Or the 7-limit diamond, because miracle
temperament works well with that too.

> Even so, your
> > calculation only seems to cover one octave. Merely to represent
> two
> > octaves with the 5:3 equivalence, you need 26 fourths.
>
> That is why I suggested beginning the second 5/3 repetion, taking it
> out to an octave, and then repeating that scale within an octave. I
> think the result is entirely practical.

What??? So if it's practical, presumably you have music made with it to
show us ...

> I take it you are using Paul's definition of 5-limit? The way I use
> the word, this doesn't mean anything, but you can for instance look at
> 3^a 5^b where a^2+ab+b^2 < 2, which I recall you doing. I don't see
> why the result cannot be used for either scale.

It can be, provided yours repeats at the octave. But that isn't a
calculation I've seen you doing.

> However, if you add 2 to the mix, you could consider 2^a 3^b 5^c with
> a^2 + b^2 + c^2 + ab + ac + bc < 2, and get a closely related linear
> programming exercise; the previous one being what one gets if we fix
> 2 to its exact value. One could certainly fix 5/3 instead. Similar
> comments apply to a least squares opitimization.

Don't know. This is looking complicated. The usual rule is that the
numerator shouldn't exceed a particular integer limit. In this case,
limits from 7 to 12 would be appropriate. You can apply that to any
generator and period you like, but you haven't applied it here, so you
don't have anything to show that your method is of value.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9oo2ic+tpk6@e...>
> Gene wrote:

> > What would you suggest
> > as a standard?

> The 11-limit tonality diamond.

If you look at the 11-limit diamond, it takes an average of 10.4
generator steps to get its elements using (7, 72) and 9.1 using
(30, 53) within 5/3; however we need to adjust that value by 72/53,
which gives 12.36; by this measure (7, 72) is better. However, where
(30, 53) really shines is in generating harmonies; we have triads in
a compass of 1, which adjusts to 1.358, compared to 13 for the
miracle system; we have 7-chords in a compass of 4, which adjusts to
5.434, compared to 13 again, and 11-chords in a compass of 6,
adjusting to 8.151, as compared to 22. The (30, 53) system is jam-
packed with harmonies, which to my mind makes it more "normal"; it
certainly helps make it practical.

> What??? So if it's practical, presumably you have music made with
it to
> show us ...

Save this sort of thing for Page Wizard or someone of that ilk, don't
try it on me. :)

You are well-equipped by knowledge and ability to evaluate this scale
without needing to write a piece of music in it, and so am I. Having
said that, I did spend yesterday working on getting myself to the
point of actually being able to produce music, and indeed plan on
writing a piece to demonstrate this scale. The relevance of that to
this discussion is more or less like the relevance of reading on
April 10, 1912 on that Thomas Andrews was going to be on the maiden
voyage of the Titanic--essentially nil, unless there is a design flaw.

You seem to have responded to a posting I canceled shortly after
posting it--I didn't like my discussion of your minimax stuff, and
was going to find the original and re-do it; however I looked through
the archives and didn't locate it.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., graham@m... wrote:
> > In-Reply-To: <9oo2ic+tpk6@e...>
> > Gene wrote:
>
> > > What would you suggest
> > > as a standard?
>
> > The 11-limit tonality diamond.
>
> If you look at the 11-limit diamond, it takes an average of 10.4
> generator steps to get its elements using (7, 72) and 9.1 using
> (30, 53) within 5/3; however we need to adjust that value by 72/53,
> which gives 12.36; by this measure (7, 72) is better. However,
where
> (30, 53) really shines is in generating harmonies; we have triads
in
> a compass of 1, which adjusts to 1.358,

I don't understand the compass or the adjustment. What kind of triads
are you talking about? How can you get three notes with only 1
iteration of a generator??

--- In tuning-math@y..., genewardsmith@j... wrote:

> However, where
> (30, 53) really shines is in generating harmonies; we have triads
in
> a compass of 1,

Oh . . . I understand this now . . . but that's cheating! Since you
don't have octave-equivalence, you're quite constrained in terms of
the inversions of the triads you can use . . . this fact can't go
unpenalized. In particular, to make nice music you'll often want to
be able to use a bass note which is the implied fundamental of the
chord.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Oh . . . I understand this now . . . but that's cheating!

It would be cheating except for the fact that I adjusted by a factor
of 72/53, which makes them directly comperable.

Since you
> don't have octave-equivalence...

But I do have octave equivalence--the proposal is to run the 4/3
around inside the 5/3 for however many times we want to do that, and
then run that pattern out to 2. We then use 2, not 5/3, as the
interval of equivalence.

--- In tuning-math@y..., genewardsmith@j... wrote:

> Since you
> > don't have octave-equivalence...
>
> But I do have octave equivalence--the proposal is to run the 4/3
> around inside the 5/3 for however many times we want to do that,
and
> then run that pattern out to 2. We then use 2, not 5/3, as the
> interval of equivalence.

Then you're breaking the pattern, and I think it's clearly
demonstrable that you have fewer of the chords you want. I'm sure
Manuel can count them easily using Scala . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:

> Then you're breaking the pattern, and I think it's clearly
> demonstrable that you have fewer of the chords you want. I'm sure
> Manuel can count them easily using Scala . . .

Scala only counted chords from one note when I tried it, so maybe I
need another tip from Manuel. However, even within the 16 notes of
5/3 of the 21 note scale, we already find lots of chords, and adding
some extra 5/3 relationships will add more.

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9oo0rt+s4tp@e...>

which scale?

> >>> temper.Temperament(441,612,temper.primes[:-2])

> 43/117, 49.0 cent generator

> So which scale do you mean?

3^(494/970) = 3^(247/485), reduced mod 3.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
>
> > Then you're breaking the pattern, and I think it's clearly
> > demonstrable that you have fewer of the chords you want. I'm sure
> > Manuel can count them easily using Scala . . .
>
> Scala only counted chords from one note when I tried it, so maybe I
> need another tip from Manuel.

I think, at least with a recent enhancement by Dave Keenan, Scala
should be able to count the chords of a given type no matter where
they appear in the scale. Manuel?

> However, even within the 16 notes of
> 5/3 of the 21 note scale, we already find lots of chords, and
adding
> some extra 5/3 relationships will add more.

Agreed. But if you count the number of 1:3:5, 1:3:7, 1:5:7, and 3:5:7
triads, and 1:3:5:7 tetrads, and their inverses, I'm betting that
Blackjack will look better than your 21-tone scale. If I'm wrong, I'm
going to have to spend a lot of time understanding why I was wrong,
which will include making color lattices for your scale, getting into
the periodicity block intepretations of it, and making Joseph compose
with it :)

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I think, at least with a recent enhancement by Dave Keenan, Scala
> should be able to count the chords of a given type no matter where
> they appear in the scale. Manuel?

I got it to count chords by means of the "location" command, and
you're right--while my scale has a lot of chords, Blackjack does even
better--for instance 8 major and 8 minor triads vs 7 major and 7
minor, 7 supermajor and subminor triads vs 6 each, and so forth.
About all my scale has going for it in the face-off is the presence
of familiar subscales such as major and minor diatonic and Paul's 10-
note PBs.

Oh well, back to the drawing board--it works, but it isn't better
than Blackjack, which I thought it would be. The ship didn't sink,
but it didn't beat the record for fastest time to New York, either.

>I think, at least with a recent enhancement by Dave Keenan, Scala
>should be able to count the chords of a given type no matter where
>they appear in the scale. Manuel?

Not a recent enhancement, use SHOW LOCATIONS. You can use integer
notation for chords, like 1:3:5:7.
I don't know Gene's scale, so can he do the comparison with Miracle
himself.

Manuel