Hypothesis revisited

Progress seems to have halted on the paper that was to introduce
MIRACLE . . .

I suggest the title

_The Relationship Between Just Intonation and Well-Formed Scales_

and some sort of "proof" of the hypothesis (I know, it doesn't always
work).

If we can do the following math problem, we'll be fine:

Given a k-by-k matrix, containing k-1 commatic unison vectors and 1
chromatic unison vector, delimiting a periodicity block, find:

(a) the generator of the resulting WF (MOS) scale;

(b) the integer N such that the interval of repetition is 1/N octaves.

If we can derive a general formula of this nature, the status of the
pathological cases (e.g., Monz' shruti block) should become clear
(hopefully). Then we can give a few examples, including the diatonic
and MIRACLE scales.

So, who's going to be our hero?

Paul wrote:

> Given a k-by-k matrix, containing k-1 commatic unison vectors and 1
> chromatic unison vector, delimiting a periodicity block, find:
>
> (a) the generator of the resulting WF (MOS) scale;

That's the bit I'm not sure about

> (b) the integer N such that the interval of repetition is 1/N octaves.

Easy. It'll usually be the determinant of the matrix. You can always get
it by solving the matrix equation. Say you have

Where H is the logs of the primes, H' is the approximation, a1...ak are
the unison vectors, where ak is chromatic, and a0 is the octave (1 0 0
... 0). You solve it to get

(a0)-1 (a0)
(a1) (a1)
H' = (a2) ( 0)H
(..) (..)
(ak) ( 0)

From which you know the first column of

(a0) (a0)-1 (a0)
(a1) (a1) (a1)
det(a2) (a2) ( 0)
(..) (..) (..)
(ak) (ak) ( 0)

will be a vector of integers specifying the number of steps to each prime
interval. You then reduce them by any common factor, and the one on top
will be the number of steps to an octave. Or say that it's pathological
if there is a common factor.

> If we can derive a general formula of this nature, the status of the
> pathological cases (e.g., Monz' shruti block) should become clear
> (hopefully). Then we can give a few examples, including the diatonic
> and MIRACLE scales.

If you supplied two different chromatic unison vectors, that would give
two equal temperaments that could be plugged into my Python script to
yield everything else we need to know.

Ideally, we could do without chromatic unison vectors altogether, but I
don't see how to do that bit. You could do a brute force search over all
consistent ETs, like my program does, but that's not the elegant way of
solving this problem.

So are we aiming for musicians or mathematicians?

Graham

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

> > (b) the integer N such that the interval of repetition is 1/N
octaves.
>
> Easy. It'll usually be the determinant of the matrix.

Huh? The determinant of the matrix is usually the number of notes,
not the number of repetitions per octave (which is usually just 1).

You can always get
> it by solving the matrix equation. Say you have
>
>
> Where H is the logs of the primes,

Looks like you left something out here, yes?

Let's leave out the octave, octave-equivalence will be assumed (yes,
in a more general case it won't be, but let's not bite off more than
we can chew).

It's fine if the paper is a bit mathematical if that helps it obtain
a more powerful result. Music theory can get very mathematical these
days.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Progress seems to have halted on the paper that was to introduce
> MIRACLE . . .
>
> I suggest the title
>
> _The Relationship Between Just Intonation and Well-Formed Scales_
>
> and some sort of "proof" of the hypothesis
...
> Then we can give a few examples, including the diatonic
> and MIRACLE scales.

Wow! This is the complete opposite direction to where I was planning
to head. To get in the mood for writing it, I was working out how to
explain to my sister, a very _practical_ violinist and strings
teacher, what was significant about the MIRACLE scales, and how a
musician can use them. Maybe JMT isn't the right place for that?

If you want to make the paper more general, then the way I'd see it
going is to list more of the best approximate JI generators as given
by Graham's program (or mine) (which would of course include meantone
diatonic for 5-limit and your decatonic for 7-limit).

You're welcome to choose particular MOS sizes for those and give the
chromatic and commatic unison vectors.

Even if you solve the problem you have proposed, how do you then
choose sets of unison vectors to give you the the _bes_ scales (small
JI errors and low cardinality for complete otonalities). Some unsison
vectors, although small, "pull in opposite directions".

The basic requirement remains the same: One algorithm for _generating_
linear temperaments and another to _filter_ out the trash. It's the
filter that is by far the most significant here, since one can, as I
did, simply feed it _every_ possible generator to a sufficiently fine
resolution.

The way I see it, the solution of your problem would merely give us
another way of generating linear temperaments to be filtered. We
already have two different ways of doing that.

Regards,
-- Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
Hi Dan, :-)

I hope the actual point of my message isn't lost because of my
ill-advised use of the word "trash". My apologies. I believe I defined
what I meant by "best" in this context, and by implication what I
meant by "trash". It's likely that "trash" isn't as emotionally loaded
a term for Australians as it is for Americans.

But hey some generators are clearly trash such as a 2 cent generator
that doesn't form a MOS until 600 notes!

Regards,
-- Dave Keenan

Paul wrote:

> Huh? The determinant of the matrix is usually the number of notes,
> not the number of repetitions per octave (which is usually just 1).

Yes, I was misteaking the terminology. The division of the octave and the
generator will come together. It's getting the number of steps to an octave
that's difficult.

> Let's leave out the octave, octave-equivalence will be assumed (yes,
> in a more general case it won't be, but let's not bite off more than
> we can chew).

No, you can't get linear temperaments without considering the octave. That
doesn't stop it being an equivalence interval. My programs assume that anyway.
To get a different equivalence interval, you re-define the coordinates.

Anyway, here's a script I worked out that prints some choices for the number of
steps of the other size. The results can be fed into my older module to get
the octave and generator. It doesn't work for the pathological case, so I
don't know what to do about that.

You'll need Numeric Python, but it can probably be adapted to whatever package
you use. Sometime I'll explain what's going on. I'm abbrevating "chromatic
unison vector" to "chroma".

import Numeric
from LinearAlgebra import inverse, determinant
from math import log

mul = Numeric.matrixmultiply

def integerize(matrix):
return Numeric.array([
[int(round(y)) for y in x]
for x in matrix])

def log2(f):
return log(f)/log(2)

primes = map(log2, [3, 5, 7, 11, 13])

for unisonVectors in (
[(-1, 2), (4, -1)],
[(0, -3), (8, 1)],
[(-1, 2), (8, 1)],
[(-1, 2), (-4, -2)],
[(0, -3), (-4, -2)],
[(0, -3, 0), (-4, -2, 0), (-2,0,-1)],
[(0, -3), (4, -1)]):

octaveSpecific = [(1,)+(0,)*len(unisonVectors)]
h = Numeric.array(primes[:len(unisonVectors)])
H = Numeric.array([1]+primes[:len(unisonVectors)])
for vector in unisonVectors:
size = mul(vector,h)
if size>0:
octave = -int(size)
else:
octave = 1-int(size)
octaveSpecific.append((octave,)+vector)

matrix = Numeric.array(octaveSpecific)
basisVectors = octaveSpecific[:2] + [(0,)*len(octaveSpecific)]*(len(octaveSpecific)-2)
basisMatrix = Numeric.array(basisVectors)

print "\n\nLeft hand defining matrix"
print matrix

octave = int(round(abs(determinant(matrix))))
inverted = inverse(matrix)*octave

print "\nH' defined by octave and chroma"
conversion = integerize([x[:2] for x in inverted])
print conversion

guess = 1/mul(octaveSpecific[1],H)

for m in range(1,int(guess*2)):
for prime in mul(conversion,(m,1)):
if prime%octave: break
else:
print m

Here are the results:

Left hand defining matrix
[[ 1 0 0]
[-3 -1 2]
[-4 4 -1]]

H' defined by octave and chroma
[[ 7 0]
[11 1]
[16 4]]
5
12
19
26

Left hand defining matrix
[[ 1 0 0]
[ 7 0 -3]
[-15 8 1]]

H' defined by octave and chroma
[[24 0]
[38 1]
[56 -8]]

Left hand defining matrix
[[ 1 0 0]
[ -3 -1 2]
[-15 8 1]]

H' defined by octave and chroma
[[17 0]
[27 -1]
[39 8]]
12
29

Left hand defining matrix
[[ 1 0 0]
[-3 -1 2]
[11 -4 -2]]

H' defined by octave and chroma
[[10 0]
[16 -2]
[23 4]]
2
12
22
32

Left hand defining matrix
[[ 1 0 0]
[ 7 0 -3]
[11 -4 -2]]

H' defined by octave and chroma
[[12 0]
[19 2]
[28 -4]]
10
22
34
46

Left hand defining matrix
[[ 1 0 0 0]
[ 7 0 -3 0]
[11 -4 -2 0]
[ 6 -2 0 -1]]

H' defined by octave and chroma
[[12 0]
[19 2]
[28 -4]
[34 -4]]
10
22
34
46

Left hand defining matrix
[[ 1 0 0]
[ 7 0 -3]
[-4 4 -1]]

H' defined by octave and chroma
[[12 0]
[19 -1]
[28 -4]]
7
19
31
43
55

Graham

"I toss therefore I am" -- Sartre

I wrote:

> Anyway, here's a script I worked out that prints some choices for the numberof
> steps of the other size. The results can be fed into my older module to get
> the octave and generator. It doesn't work for the pathological case, so I
> don't know what to do about that.

I've fixed that, and thrown a quick web page together at
<http://x31eq.com/vectors.html>. I'll try and make it easier to
throw lots of sets of unison vectors at it now. If anybody has some favourites
they'd like me to try, send them in!

Graham

"I toss therefore I am" -- Sartre

> From: Graham Breed <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, June 23, 2001 6:04 AM
> Subject: [tuning-math] Re: Hypothesis revisited
>
>
> ... I'm abbrevating "chromatic unison vector" to "chroma".

Hmmm... that's really interesting.

"Chroma" is a music-theory term with quite a background history.

(Maybe Paul will say more.)

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

After getting the computer to chuck out this kind of thing:

> H' defined by octave and chroma
> [[ 7 0]
> [11 1]
> [16 4]]
> 5
> 12
> 19
> 26

I've now got some idea what it means.

-- A complete set of unison vectors gives an equal temperament

-- Take one of them away, you get a linear temperament.

This can be thought of as a section of the scale tree. The
usual fifth based scales are

7 5
12
19 17
26 31 29 22

And the list here is a subset of that branch.

In particular, though, it's Erv Wilson's septimally positive set. This is true
forever, even when the temperaments stop being consistent. They are the EDOs
with 5+7n notes in them.

-- Put it back as a chromatic UV, and you get an n-ly mth-ly positive set

In general, the n in the n-ly bit is the number of notes in the equal
temperament you got at the first step. This makes sense: the more notes you
add in a Wilson/Bosanquet pattern, the closer you get to that temperament. So,
in the example above, it gets closer and closer to 7-equal *the smaller the
chromatic unison vector gets*. When that unison vector becomes a unison, and
so commatic, you do have the equal temperament.

The amount of positivity is less obvious. It assumes some kind of
fifth generators, and so isn't that general. So really it's the "5" that's
important for septimally positive scales rather than the "+1".

Graham

Graham

"I toss therefore I am" -- Sartre

Monz wrote:

> Hmmm... that's really interesting.
>
> "Chroma" is a music-theory term with quite a background history.

Oh, well, see if you can come up with a better word for "chromatic unison
vector" for when I start explaining this.

Graham

"I toss therefore I am" -- Sartre

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning-math/message/298

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Progress seems to have halted on the paper that was to introduce
> > MIRACLE . . .
> >
> > I suggest the title
> >
> > _The Relationship Between Just Intonation and Well-Formed Scales_
> >
> > and some sort of "proof" of the hypothesis
> ...
> > Then we can give a few examples, including the diatonic
> > and MIRACLE scales.
>
> Wow! This is the complete opposite direction to where I was
planning
> to head. To get in the mood for writing it, I was working out how
to
> explain to my sister, a very _practical_ violinist and strings
> teacher, what was significant about the MIRACLE scales, and how a
> musician can use them.

That would be a great article... I would love to read *that* one!

_______ ______ ______
Joseph Pehrson

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

/tuning-math/message/300

> I think my situation is perhaps made more difficult than some others
> around here in that I actually enjoy and have a real interest in the
> theory end of this subject... and while I do respect what I've
learned
> from that "small handful of likeminded folks", I can't help but at
> times like this feeling that were I hostile or ambivalent towards it
> all everything sure would be a hell of a lot easier.
>

It certainly would... because it would be a *lot* easier to "dismiss"
your posts! Well, of all the people around here, with the exception
of possibly Margo Schulter, you have shown Dan, that one can make
music out of just about *any* possible scales... the "good," bad and
the ugly... the so-called "ugly" sometimes becoming quite beautiful...

It seems this attitude is also shared to some extent by Brian
McLaren... who has obviously had a lot of microtonal listening
experience...

Frankly, I'm fascinated with the "special properties" of scales such
as MIRACLE, but truly you've convinced me that there are "other
things around..."

________ ______ _______
Joseph Pehrson

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

/tuning-math/message/305

>
> > From: Graham Breed <graham@m...>
> > To: <tuning-math@y...>
> > Sent: Saturday, June 23, 2001 6:04 AM
> > Subject: [tuning-math] Re: Hypothesis revisited
> >
> >
> > ... I'm abbrevating "chromatic unison vector" to "chroma".
>
>
> Hmmm... that's really interesting.
>
> "Chroma" is a music-theory term with quite a background history.
>
> (Maybe Paul will say more.)
>

What is the history of that term again, Monz??

Joe P.

> ----- Original Message -----
> From: Graham Breed <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, June 23, 2001 11:41 AM
> Subject: [tuning-math] Re: Hypothesis revisited
>
>
> Monz wrote:
>
> > Hmmm... that's really interesting.
> >
> > "Chroma" is a music-theory term with quite a background history.
>
> Oh, well, see if you can come up with a better word for "chromatic unison
> vector" for when I start explaining this.

I'm sorry, Graham... perhaps I should have been clear when I wrote
that, that this might be an appropriate *new* additional definition of
the term "chroma".

Perhaps, based on the wide background history I mention, your
new use of "chroma" fits right in. I'd have to dig out what I have
about this term and can't do it right now.

Maybe until then, you could use "chromuv"?

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> ----- Original Message -----
> From: <jpehrson@rcn.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, June 23, 2001 8:11 PM
> Subject: [tuning-math] Re: Hypothesis revisited
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> /tuning-math/message/305
>
> > Hmmm... that's really interesting.
> >
> > "Chroma" is a music-theory term with quite a background history.
> >
> > (Maybe Paul will say more.)
> >
>
> What is the history of that term again, Monz??

Joe, I'd have to shift gears in my mind and do some real
research to answer this question in the detail it deserves.

You (and others who have my book) can read a few references
to "chroma" in the beginning chapters.

In brief, "chroma" is a Greek term referring to color.

It has been used by later music-theorists primarily to
refer to the idea of a categorical pitch-class _gestalt_.

Some theorists have used it very similarly to the way
I have... that is, to refer to the difficult-to-describe
"affect" produced by the basic prime intervals (i.e.,
2:1, 3:2, 5:4, 7:4, 11:8, 13:8, etc.).

But IIRC Paul has criticized me for using it this way
because it had various other meanings in the past which
may be more well-established. Paul, help!

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Even if you solve the problem you have proposed, how do you
then
> choose sets of unison vectors to give you the the _bes_ scales
(small
> JI errors and low cardinality for complete otonalities). Some
unsison
> vectors, although small, "pull in opposite directions".

One way would be to examine the geometry of the unison vectors
in the triangular lattice -- if the angles between them are small,
the periodicity block will not contain a lot of consonant structures
. . .
>
> The basic requirement remains the same: One algorithm for
_generating_
> linear temperaments and another to _filter_ out the trash. It's
the
> filter that is by far the most significant here, since one can, as I
> did, simply feed it _every_ possible generator to a sufficiently
fine
> resolution.

Who's to say what's trash?
>
> The way I see it, the solution of your problem would merely give
us
> another way of generating linear temperaments to be filtered.
We
> already have two different ways of doing that.
>
But the most significant part of the paper, I believe, would be to
show how well-formed scales, which have received a great deal
of attention in the music-theoretic literature of late, can be seen
as flowing naturally from a fundametally JI-oriented framework,
which has received virtually none.

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
>
> > Let's leave out the octave, octave-equivalence will be assumed
(yes,
> > in a more general case it won't be, but let's not bite off more
than
> > we can chew).
>
> No, you can't get linear temperaments without considering the
octave. That
> doesn't stop it being an equivalence interval. My programs assume
that anyway.

Then what do you mean, "No"? What I'm thinking is, let's not bother
with a column for powers of 2 in the matrices . . . along the lines
of what Fokker did.

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> After getting the computer to chuck out this kind of thing:
>
> > H' defined by octave and chroma
> > [[ 7 0]
> > [11 1]
> > [16 4]]
> > 5
> > 12
> > 19
> > 26
>
> I've now got some idea what it means.
>
> -- A complete set of unison vectors gives an equal temperament

If you temper them all out.
>
> -- Take one of them away, you get a linear temperament.

Yes, and reasonable choices of an additional, non-tempered-out unison
vector lead to MOSs of the linear temperament.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Even if you solve the problem you have proposed, how do you
> then
> > choose sets of unison vectors to give you the the _bes_ scales
> (small
> > JI errors and low cardinality for complete otonalities). Some
> unsison
> > vectors, although small, "pull in opposite directions".
>
> One way would be to examine the geometry of the unison vectors
> in the triangular lattice -- if the angles between them are small,
> the periodicity block will not contain a lot of consonant structures
> . . .

But doesn't that depend which set of unison vectors you use for a
given PB, since they are not unique.

I think of Canasta having many consonances because 224:225 and 385:384
have such a _small_ angle between them when projected onto the 5-limit
plane.

But yes, I'm sure you could do something like this, but why bother,
when we have a "near-JI filter" on the end of the pipeline.

> > The basic requirement remains the same: One algorithm for
> _generating_
> > linear temperaments and another to _filter_ out the trash. It's
> the
> > filter that is by far the most significant here, since one can, as
I
> > did, simply feed it _every_ possible generator to a sufficiently
> fine
> > resolution.
>
> Who's to say what's trash?

This rhetorical question only serves to strengthen my argument that it
is the _filter_ that is most significant.

If the question is not quite rhetorical:
1. No one thinks that all linear temperaments are equally interesting.
2. Masses of people over centuries have effectively given us a short
list of those they found useful. (Popularity of Partch's scales would
in effect tell us that MIRACLE is useful)
3. There is wide acceptance (even by Dan Stearns :-) that
approximation of small whole-number ratios contributes _something_
towards making a linear temperament useful. It's certainly one of
_your_ key assumptions Paul. It's presumably the reason why you're
interested in unison vectors in the first place.

> > The way I see it, the solution of your problem would merely give
> us
> > another way of generating linear temperaments to be filtered.
> We
> > already have two different ways of doing that.
> >
> But the most significant part of the paper, I believe, would be to
> show how well-formed scales, which have received a great deal
> of attention in the music-theoretic literature of late, can be seen
> as flowing naturally from a fundametally JI-oriented framework,
> which has received virtually none.

But they don't "flow naturally", do they? What is the definition of
"well-formedness"? Is it simply MOS/Myhill's?

But presumably all you want to do is show that the current definition
of "well-formedness" isn't enough, and that additional criteria of
near-JI-ness should be added. To do this, one can show the
near-JI-ness of some historical scales. (Of course some are not). Then
you can generate well formed scales that have no approximations of
SWNRs and let people decide whether they find them useful. A maximally
dissonant MOS, there's a fun project. :-)

All this scan be done without having to mention periodicity blocks or
unison vectors at all.

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> >
> > One way would be to examine the geometry of the unison
vectors
> > in the triangular lattice -- if the angles between them are
small,
> > the periodicity block will not contain a lot of consonant
structures
> > . . .
>
> But doesn't that depend which set of unison vectors you use for
a
> given PB, since they are not unique.

Well, the picture is not that simple when you're talking about one
of the unison vectors (the chromatic one) _not_ being tempered
out. Then it _does_ matter which set you choose.
>
> I think of Canasta having many consonances because
224:225 and 385:384
> have such a _small_ angle between them when projected onto
the 5-limit
> plane.

Hmm . . . can you explain the thinking behind that? Of course, the
fact that you're projecting them makes it very different from the
idea I was thinking about.
>
> But yes, I'm sure you could do something like this, but why
bother,
> when we have a "near-JI filter" on the end of the pipeline.

To make the process more transparent and intuitive for those
who prefer to look at, and work with, JI lattices.
>
>
> If the question is not quite rhetorical:
> 1. No one thinks that all linear temperaments are equally
interesting.

Erv Wilson?

> 2. Masses of people over centuries have effectively given us a
short
> list of those they found useful. (Popularity of Partch's scales
would
> in effect tell us that MIRACLE is useful)

wha . . . wha . . . what??

> 3. There is wide acceptance (even by Dan Stearns :-) that
> approximation of small whole-number ratios contributes
_something_
> towards making a linear temperament useful. It's certainly one
of
> _your_ key assumptions Paul. It's presumably the reason why
you're
> interested in unison vectors in the first place.

Yup! I just thought this paper would be better if it were capable of
unifying different fields of tuning theory, and presenting a few
new interesting scales with descriptions according to this new
unified theory, than being some sort of attempt to crown a few
scales with the title of "best". Of course, mentioning these
searches would be very relevant in the context of the paper, but I
see it as more of a footnote than as the main subject of the
paper. Every scale has its unique properties, so ruling out any
just because others are "better" means blocking off many
potentially interesting musical effects.
>
> > But the most significant part of the paper, I believe, would be
to
> > show how well-formed scales, which have received a great
deal
> > of attention in the music-theoretic literature of late, can be
seen
> > as flowing naturally from a fundametally JI-oriented
framework,
> > which has received virtually none.
>
> But they don't "flow naturally", do they? What is the definition of
> "well-formedness"? Is it simply MOS/Myhill's?

Yes. So why don't they "flow naturally"?
>
> But presumably all you want to do is show that the current
definition
> of "well-formedness" isn't enough, and that additional criteria
of
> near-JI-ness should be added.

More than that -- I want to show that well-formedness should not
be an "axiom" at all but could instead be derived from more
"fundamental considerations". A JI-friendly underpinning to much
modern scale theory. One might even include a case where
_two_ of the unison vectors are not tempered out, and related
this to a second-order ME scale, such as the Indian 7-out-of-22.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > One way would be to examine the geometry of the unison
> vectors
> > > in the triangular lattice -- if the angles between them are
> small,
> > > the periodicity block will not contain a lot of consonant
> structures
> > > . . .
> >
> > But doesn't that depend which set of unison vectors you use for
> a
> > given PB, since they are not unique.
>
> Well, the picture is not that simple when you're talking about one
> of the unison vectors (the chromatic one) _not_ being tempered
> out. Then it _does_ matter which set you choose.

Yes. That's what I thought I said. It _does_ matter. But choosing one
to be chromatic, still doesn't uniquely determine the others does it?
(except in 5-limit). So how do you know which vectors to check angles
between?

> > I think of Canasta having many consonances because
> 224:225 and 385:384
> > have such a _small_ angle between them when projected onto
> the 5-limit
> > plane.
>
> Hmm . . . can you explain the thinking behind that? Of course, the
> fact that you're projecting them makes it very different from the
> idea I was thinking about.

Yes. A different idea. This was how I found Canasta's
planar-temperament precursor. I started at a note in the 5-limit
lattice and I knew if I grew the scale in a particular approximate
direction (pair of opposing directions) I would get both aproximate
7's and approximate 11's.

> > But yes, I'm sure you could do something like this, but why
> bother,
> > when we have a "near-JI filter" on the end of the pipeline.
>
> To make the process more transparent and intuitive for those
> who prefer to look at, and work with, JI lattices.

That's a worthy aim, but it can be acheived by finding the
linear-temperaments by existing methods and working backwards to the
unison vectors.

> > If the question is not quite rhetorical:
> > 1. No one thinks that all linear temperaments are equally
> interesting.
>
> Erv Wilson?

Doesn't it seem to you, from his diagrams, that he at least considers
noble generators to be more interesting or useful or special in some
way? He also obviously believes, as we do, that SWNRs (and nearby) are
special.

> > 2. Masses of people over centuries have effectively given us a
> short
> > list of those they found useful. (Popularity of Partch's scales
> would
> > in effect tell us that MIRACLE is useful)
>
> wha . . . wha . . . what??

I assume you're not objecting to the first sentence? I'll adress the
second. Graham Breed (and George Secor) have shown that MIRACLE_41 is
almost identical to several of Partch's scales. I can't help seeing
Partch's various scales as gropings towards either Canasta or
MIRACLE-41. I expect Partch would not have been able to distinguish
his scales from the corresponding MIRACLE-temperament of them, since I
understand someone said he couldn't distinguish one of them from
41-EDO. I think the fact that Partch, doing it mostly by ear, and we,
doing it mostly by math, (and George Secor doing it by ???),
essentially converged on the same thing, is no accident.

> > 3. There is wide acceptance (even by Dan Stearns :-) that
> > approximation of small whole-number ratios contributes
> _something_
> > towards making a linear temperament useful. It's certainly one
> of
> > _your_ key assumptions Paul. It's presumably the reason why
> you're
> > interested in unison vectors in the first place.
>
> Yup! I just thought this paper would be better if it were capable of
> unifying different fields of tuning theory, and presenting a few
> new interesting scales with descriptions according to this new
> unified theory, than being some sort of attempt to crown a few
> scales with the title of "best".

Gimme a break Paul. Dan's already slapped me on the wrist for that.
The "political correctness police" are getting a little tedious.

I thought I made it clear that by "best" I wasn't trying to claim
something which is _obviously_ a matter of personal taste. I first set
up some criteria (which incidentally an awful lot of people find,
align well to their personal taste, at least some of the time) and
then I talk about what is "best" according to those criteria. Surely I
don't have to re-state these criteria in every post I make, especially
when it's to the tuning-math list?

> Of course, mentioning these
> searches would be very relevant in the context of the paper, but I
> see it as more of a footnote than as the main subject of the
> paper.

Me too. Just enought to say that we ran these searches and we found
the previouly mentioned scales (which you are welcome to introduce in
the manner of "Forms of Tonality" using unison vectors) to be the
"best".

> Every scale has its unique properties, so ruling out any
> just because others are "better" means blocking off many
> potentially interesting musical effects.

Who is ruling out such scales. You mean you don't think I should have
ruled out a MOS with a 2 cent generator. Oh. Well sorry.

> > But they don't "flow naturally", do they? What is the definition
of
> > "well-formedness"? Is it simply MOS/Myhill's?
>
> Yes. So why don't they "flow naturally"?

Because there are zillions of MOS scales that have no relationship
with small unison vectors. Sure you could probably always find a
corresponding periodicity block, but these will have "unison vectors"
so large as not to merit the name.

> > But presumably all you want to do is show that the current
> definition
> > of "well-formedness" isn't enough, and that additional criteria
> of
> > near-JI-ness should be added.
>
> More than that -- I want to show that well-formedness should not
> be an "axiom" at all but could instead be derived from more
> "fundamental considerations". A JI-friendly underpinning to much
> modern scale theory. One might even include a case where
> _two_ of the unison vectors are not tempered out, and related
> this to a second-order ME scale, such as the Indian 7-out-of-22.

See response to previous paragraph. You can't derive MOS from JI or
vice versa. One is a horizontal melodic property, the other vertical
harmonic. Periodicity blocks may give you MOS approx-JI scales but
they won't give you the MOS non-approx-JI scales.

Regards,
-- Dave Keenan

I wrote:
> Because there are zillions of MOS scales that have no relationship
> with small unison vectors. Sure you could probably always find a
> corresponding periodicity block, but these will have "unison
vectors"
> so large as not to merit the name.

Try this one: A chain of 10, 369c generators, octave period.

> almost identical to several of Partch's scales. I can't help seeing
> Partch's various scales as gropings towards either Canasta or
> MIRACLE-41. I expect Partch would not have been able to distinguish
> his scales from the corresponding MIRACLE-temperament of them,
since I
> understand someone said he couldn't distinguish one of them from
> 41-EDO. I think the fact that Partch, doing it mostly by ear, and
we,
> doing it mostly by math, (and George Secor doing it by ???),
> essentially converged on the same thing, is no accident.
>

I'm getting a little confused here... Did Harry Partch use a 41-tone
scale in addition to his 43-tone scale?? He never actually
used "Miracle 41" did he??

_________ _______ _______
Joseph Pehrson

Hmmm ... nowhere in Genesis of a Music do I see any reference to 41 EDO.
There are references to 19 EDO and 53 EDO IIRC, but I don't remember any
41s. As far as I can tell, Partch started with the 28 tonalities -- 12
primary and 16 secondary. That makes a 29-note scale from 1/1 through 2/1.
Then he filled in some of the larger the gaps in this scale with notes from
the secondary tonalities. It's never been clear to me why he stopped at 43,
though, rather than completing all 28 tonalities.
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

> -----Original Message-----
> From: jpehrson@rcn.com [mailto:jpehrson@rcn.com]
> Sent: Sunday, June 24, 2001 8:25 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] 41 "miracle" and 43 tone scales
>
>
>
> > almost identical to several of Partch's scales. I can't help seeing
> > Partch's various scales as gropings towards either Canasta or
> > MIRACLE-41. I expect Partch would not have been able to distinguish
> > his scales from the corresponding MIRACLE-temperament of them,
> since I
> > understand someone said he couldn't distinguish one of them from
> > 41-EDO. I think the fact that Partch, doing it mostly by ear, and
> we,
> > doing it mostly by math, (and George Secor doing it by ???),
> > essentially converged on the same thing, is no accident.
> >
>
> I'm getting a little confused here... Did Harry Partch use a 41-tone
> scale in addition to his 43-tone scale?? He never actually
> used "Miracle 41" did he??
>
> _________ _______ _______
> Joseph Pehrson
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

--- In tuning-math@y..., jpehrson@r... wrote:
>
> > almost identical to several of Partch's scales. I can't help
seeing
> > Partch's various scales as gropings towards either Canasta or
> > MIRACLE-41. I expect Partch would not have been able to
distinguish
> > his scales from the corresponding MIRACLE-temperament of them,
> since I
> > understand someone said he couldn't distinguish one of them from
> > 41-EDO. I think the fact that Partch, doing it mostly by ear, and
> we,
> > doing it mostly by math, (and George Secor doing it by ???),
> > essentially converged on the same thing, is no accident.
> >
>
> I'm getting a little confused here... Did Harry Partch use a
41-tone
> scale in addition to his 43-tone scale?? He never actually
> used "Miracle 41" did he??
>
> _________ _______ _______
> Joseph Pehrson

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> I wrote:
> > Because there are zillions of MOS scales that have no relationship
> > with small unison vectors. Sure you could probably always find a
> > corresponding periodicity block, but these will have "unison
> vectors"
> > so large as not to merit the name.
>
> Try this one: A chain of 10, 369c generators, octave period.

Sorry. That should have been "A 10 note chain of 369c generators..."

Sorry about the previous message, sent by mistake.

--- In tuning-math@y..., jpehrson@r... wrote:
> I'm getting a little confused here... Did Harry Partch use a
> 41-tone scale in addition to his 43-tone scale??

Sure. There are two in the Scala archive, But that's not what I meant.

> He never actually used "Miracle 41" did he??

No. I didn't say that either. But he might not have noticed if someone
had substituted a scale which was MIRACLE-41 plus a couple of extra
notes from MIRACLE-45.

Read:
http://www.anaphoria.com/secor.PDF
and
http://x31eq.com/decimal_lattice.htm#partch
and then tell me what you don't understand.

Regards,
-- Dave Keenan

> ----- Original Message -----
> From: M. Edward Borasky <znmeb@aracnet.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, June 24, 2001 8:42 PM
> Subject: RE: [tuning-math] 41 "miracle" and 43 tone scales
>
>
> Hmmm ... nowhere in Genesis of a Music do I see any reference to 41 EDO.
> There are references to 19 EDO and 53 EDO IIRC, but I don't remember any
> 41s.

Hi Ed. Other than 12-EDO, the others Partch discusses are 19, 36,
and 53, in connection with Yasser, Busoni, and Mercator, respectively.
He never hints that he would consider using any of them himself.
And you're right, he says nothing about 41.

It was Erv Wilson who hypothesized that Partch was intuitively
"feeling out" a version of 41-EDO where two of the pitches could
imply either of a pair of ratios (12/11 and 11/10, and their
"octave"-complements).

> As far as I can tell, Partch started with the 28 tonalities -- 12
> primary and 16 secondary. That makes a 29-note scale from 1/1 through 2/1.
> Then he filled in some of the larger the gaps in this scale with notes
from
> the secondary tonalities. It's never been clear to me why he stopped at
43,
> though, rather than completing all 28 tonalities.

The 29-tone scale comes directly from the 11-limit Tonality Diamond,
and only involves the secondary tonalities in that they are
*partially* present within that scale.

Partch got those 29 pitches from the 12 primary tonalities:
6 otonal and 6 utonal hexads. Because 1/1 is represented 5 times,
and 4/3 and 3/2 each represented twice, the potential (6*6) = 36
different pitches are reduced to 36 - 5 - 2 = 29.

Partch was essentially satisfied with the harmonic possibilites
of this 29-tone scale, since the formed his neat and compact
Tonality Diamond. He "filled in the gaps" mainly because he
wanted a certain measure of melodic evenness in the basic scale
which formed essentially his full set of resources.

(I qualify this with "essentially" because there are many,
many other ratios which do in fact appear in Partch's compositions.
As he himself emphatically reiterated, he considered the 43
pitches to be only a peripheral aspect of his whole technique,
and it was a limitation which he often ignored.)

Once he reached the point where the whole 2:1 was divided into
approximately equal steps, he stopped. That division happened
to be into 43 different degrees.

You're correct that the notes filling the gaps were taken
from expansion of the pitch-space into the secondary
tonalities, so that the new pitches would form familiar
harmonic relationships to the primary ones. But Partch's
main consideration in choosing the new pitches was to divide
the melodic gaps in the scale into the appropriately-spaced
intervals in terms of *pitch-height*.

So his goal was not to complete the secondary tonalities.
If he had chosen more than the 14 secondary pitches he
did choose, he would have ended up melodically with either
less even spacing throughout one or more of the gaps,
or only some of 43-tone steps divided in half and others
not divided, which would give a scale still less even.

And in definite answer to Joe Pehrson's question: NO,
Partch *never* considered MIRACLE or any other temperament.

One Partch made the break with 12-EDO around 1929 or so,
he never wrote any other music in non-JI tunings, with
the sole exception of the piano parts of _Bitter Music_
in the mid-1930s, and which "piece" was really a private
journal and which he thought he had destroyed before he
died. (_Bitter Music_ only exists now because a copy was
stored on microfilm at a university and got past Partch.)

*We* (Joe P., myself, Dave Keenan, Paul, Graham, Herman, and
the others interested in MIRACLE) are the ones who like it's
terrific emulation of Partch's scale.

Hmmm... but George Secor knew Partch too. I wonder if
Partch was familiar with Secor's discovery of MIRACLE...?

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave K.,
>
> <<Try this one: A chain of 10, 369c generators, octave period.>>
>
> Though this sounds more like a loaded mousetrap than a practical
type
> question... if small is all your really looking for, how about
> something on the order of 49/40 and 4375/4096?
>
> --Dan Stearns

Actually, with the 10-note 369c MOS, I was looking for a MOS scale
that Paul would have difficulty finding unison-vectors for, that are
anything like unisons. i.e. This one was meant to have _big_ UVs, and
not to contain any good approximations to SWNRs.

Are you asking us to find a linear temperament that treats those
unison vectors (49/40 and 4375/4096) as commas, and to tell you how
"good" it is relative to the usual JI criteria.

I don't know how. But Graham or Paul may be able to soon.

In-Reply-To: <021301c0fd42$4989a440$77bcd33f@stearns>
Dan Stearns wrote:

> I think it was in Blackwood's book, I don't have right now so I can't
> check, that I remember the 25/24 being called a "minor chroma" and the
> 135/128 a "major chroma".

I did get hold of that book a while back, so it may be where I got the
idea from. (I say "may be" because I don't consciously remember this,
but as I read the book not long ago I can't claim it's a coincidence.)

> So generalizing commatic unison vectors in periodicity blocks as
> chromas would seem at odds with this as the 25/24 "minor chroma" is a
> chromatic unison vector in the two-dimensional diatonic periodicity
> block.

Sure, but it's the *chromatic* unison vector I was going to call a
"chroma" so no problem. The *commatic* unison vectors can easily enough
be called "commas".

> (Incidentally, I think the 135/128 "major chroma" is a
> chromatic unison vector of the so-called miracle generator at
> two-dimensions; with 34171875/33554432 being the commatic unison
> vector if the generator is taken to a 10- or 11-tone MOS.)

Don't know about this offhand. When I get home, I might plug it in.

Graham

In-Reply-To: <004601c0fd40$bfed2d20$4448620c@att.com>
monz wrote:

> Hmmm... but George Secor knew Partch too. I wonder if
> Partch was familiar with Secor's discovery of MIRACLE...?

Without hard facts, all we have is speculation. Which is good, because
it's much more fun that way.

If Secor had shared this with Partch, I'm surprised he didn't find out
about the earlier 43 note scale that fits Miracle better. I suspect if
he knew Partch, he would also have known Wilson, hence learned of the 41-
connection from him?

One question is, how much did Partch know about Miracle when he drew up
that original, unpublished scale? It may be stretching credulity to
suggest he worked it all out, and then pretended it was pure JI. But the
criteria he was using may well have matched those that are enshrined in
Miracle. Roughly equal melodic steps will of course favour an MOS. And
he would have been able to hear the intervals that were almost just by
Miracle approximations. And so he could have chosen the extra notes to
maximise these consonances.

In which case, why did he change his mind later? I think it was to get
more modulation by fifths in the 5-limit plane. With experience, he
decided this was more important than matching the consonances.

The limitations on modulation by fifths is one of the problems with
Miracle, at least in a traditional context. Boomsliter and Creel's
theories work very well with schismic, but not at all well with Miracle,
temperament.

Graham

In-Reply-To: <9h68ft+386j@eGroups.com>
Dave Keenan wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Well, the picture is not that simple when you're talking about one
> > of the unison vectors (the chromatic one) _not_ being tempered
> > out. Then it _does_ matter which set you choose.
>
> Yes. That's what I thought I said. It _does_ matter. But choosing one
> to be chromatic, still doesn't uniquely determine the others does it?
> (except in 5-limit). So how do you know which vectors to check angles
> between?

Yes, it does matter, but the vectors aren't unique. Where you have more
than one commatic vector you have a lot of freedom about which you
choose. If you check the output file from my latest script, you should
see this twice:

mapping by steps:
[[10 1]
[16 1]
[23 3]
[28 3]
[35 2]]

It means there are two ways of defining 10 (or 1+10n) note Miracle.
However, that may not be a good example because one of them is
pathological: it actually gives a 20 note periodicity block, which is
why I included it in the test. But it's the *chromatic* vector that
differs, so there is more than one that works.

The unison vectors I used for 31+41n are:

[[ 2 -2 2 0 -1]
[-7 -1 1 1 1]
[-1 5 0 0 -2]
[-5 2 2 -1 0]]

That uses 100:99 as the chromatic UV. The more obvious choice would be a
schisma, so that

[[-15 8 1 0 0]
[-7 -1 1 1 1]
[-1 5 0 0 -2]
[-5 2 2 -1 0]]

would give the same results. I can't check this now, as I don't have
Numerical Python installed, or even Excel. But you may be able to. Try
inverting this matrix, and multiplying it by its determinant:

[[ 1 0 0 0 0]
[-15 8 1 0 0]
[-7 -1 1 1 1]
[-1 5 0 0 -2]
[-5 2 2 -1 0]]

The left hand two columns should be

[[ 41 0]
[ 65 -6]
[ 95 7]
[115 2]
[142 -15]]

If they are, the two sets of unison vectors give exactly the same
results. I think they must be, because I remember checking the
determinant before, and any chroma that gives a determinant of 41 when
placed with Miracle commas should give this result.

The original matrix has an octave as the top row, the chroma as the next
one down, and commas below that. Inverting it and taking the left hand
two columns defines the prime intervals in terms of the octave and
chroma. If the left hand column has a common factor, divide through by
that factor. If the right hand column has a prime factor, that tells you
how many equal parts you need to divide the octave into, but you don't
need to worry about that yet.

To get the MOS, you need to add a multiple of the left hand column to the
right hand column so that it's divisible by the number of steps to the
octave. This is what my program does. Use this as a new right-hand
column and you have defined the octave in terms of two step sizes.

You most certainly do need octave-specific matrices. Otherwise, that
left-hand column won't be there. You also need to make sure the chroma
is a small interval. There may be an algorithm that works with octave
invariant matrices, but it's easier to upgrade them to be
octave-specific, and use a common or garden inverse.

Graham

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Well, the picture is not that simple when you're talking about one
> > of the unison vectors (the chromatic one) _not_ being tempered
> > out. Then it _does_ matter which set you choose.
>
> Yes. That's what I thought I said. It _does_ matter. But choosing one
> to be chromatic, still doesn't uniquely determine the others does it?
> (except in 5-limit). So how do you know which vectors to check angles
> between?

You're right . . . the angle stuff only makes sense if two or more unison vectors are not being
tempered out.

> That's a worthy aim, but it can be acheived by finding the
> linear-temperaments by existing methods and working backwards to the
> unison vectors.

Correct.
>
> > > If the question is not quite rhetorical:
> > > 1. No one thinks that all linear temperaments are equally
> > interesting.
> >
> > Erv Wilson?
>
> Doesn't it seem to you, from his diagrams, that he at least considers
> noble generators to be more interesting or useful or special in some
> way?

We all know the special properties of noble generators . . . as far as Wilson being exclusive
about them, Kraig has reported otherwise . . .
>
> > > 2. Masses of people over centuries have effectively given us a
> > short
> > > list of those they found useful. (Popularity of Partch's scales
> > would
> > > in effect tell us that MIRACLE is useful)
> >
> > wha . . . wha . . . what??
>
> I assume you're not objecting to the first sentence?

I am.

> I'll adress the
> second. Graham Breed (and George Secor) have shown that MIRACLE_41 is
> almost identical to several of Partch's scales.

Eh . . . not quite.

> I can't help seeing
> Partch's various scales as gropings towards either Canasta

Don't see it.

> or
> MIRACLE-41.

Toward modulus-41, yes . . . with many other generators functioning as well as, if not better than,
the 4/41 (MIRACLE) generator.

> > Yup! I just thought this paper would be better if it were capable of
> > unifying different fields of tuning theory, and presenting a few
> > new interesting scales with descriptions according to this new
> > unified theory, than being some sort of attempt to crown a few
> > scales with the title of "best".
>
> Gimme a break Paul. Dan's already slapped me on the wrist for that.
> The "political correctness police" are getting a little tedious.

I didn't see Dan's post on this, and believe me, the last thing I want to do is be politically correct.
> >
> > Yes. So why don't they "flow naturally"?
>
> Because there are zillions of MOS scales that have no relationship
> with small unison vectors.

But the _whole idea_ of MOS -- where does that come from? Really just from looking at the
diatonic scale and then generalizing. So perhaps I'm interested in showing _why_ the diatonic
scale is MOS, and giving an _impetus_ for finding more MOSs . . . without taking it as an axiom
that MOSs are special.

> > More than that -- I want to show that well-formedness should not
> > be an "axiom" at all but could instead be derived from more
> > "fundamental considerations". A JI-friendly underpinning to much
> > modern scale theory. One might even include a case where
> > _two_ of the unison vectors are not tempered out, and related
> > this to a second-order ME scale, such as the Indian 7-out-of-22.
>
> See response to previous paragraph. You can't derive MOS from JI or
> vice versa. One is a horizontal melodic property, the other vertical
> harmonic. Periodicity blocks may give you MOS approx-JI scales but
> they won't give you the MOS non-approx-JI scales.

See above. Yes, Dave, we both want to "rule out" the MOSs with no approximations to any JI
intervals/chords (if such a thing is possible). That is where we (the originators of "MIRACLE")
differ from Dan Stearns (at least in the viewpoint that goes behind this paper we're
contemplating). But that still leaves a great number of possibilities, as Robert Valentine, for
example, has been finding.

--- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:
> Hmmm ... nowhere in Genesis of a Music do I see any reference to 41 EDO.
> There are references to 19 EDO and 53 EDO IIRC, but I don't remember any
> 41s.

Genesis of a Music was written very early in Partch's career. Later, he met Erv Wilson, who
played Partch 41-tET and Partch couldn't distinguish it from his scale.

> As far as I can tell, Partch started with the 28 tonalities -- 12
> primary and 16 secondary. That makes a 29-note scale from 1/1 through 2/1.

The original 29-note scale is simply the Diamond -- only the primary 11-limit ratios. It has to do
only with 6 Otonalities and 6 Utonalities, all containing 1/1. No secondary tonalities are explicitly
involved at this stage.

> Then he filled in some of the larger the gaps in this scale with notes from
> the secondary tonalities. It's never been clear to me why he stopped at 43,
> though, rather than completing all 28 tonalities.

He stopped at 43 in order to make a melodically fairly even scale. With 10/9 and 11/10 seen as
a commatic pair (the unison vector involved is 100:99), and their octave complements another
such pair, Partch's scale is a 41-tone periodicity block -- or what Wilson calls a "Constant
Structure".

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Joe,
>
> I think it was in Blackwood's book, I don't have right now so I can't
> check, that I remember the 25/24 being called a "minor chroma" and the
> 135/128 a "major chroma".
>
> So generalizing commatic unison vectors in periodicity blocks as
> chromas would seem at odds with this as the 25/24 "minor chroma" is a
> chromatic unison vector in the two-dimensional diatonic periodicity
> block.

No one was proposing generalizing commatic unison vectors as "chromas". They were
suggesting generalizing _chromatic_ unison vectors as "chromas".

> (Incidentally, I think the 135/128 "major chroma" is a
> chromatic unison vector of the so-called miracle generator at
> two-dimensions; with 34171875/33554432 being the commatic unison
> vector if the generator is taken to a 10- or 11-tone MOS.)

I see the MIRACLE scales as needing three or four unison vectors each, since they live in a 7-
or 11-limit lattice (i.e., they're 3D or 4D).

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> It was Erv Wilson who hypothesized that Partch was intuitively
> "feeling out" a version of 41-EDO where two of the pitches could
> imply either of a pair of ratios (12/11 and 11/10, and their
> "octave"-complements).

Actually, the pair was 11/10 and 10/9 . . . you don't get a PB or CS the other way.

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Hi Dave K.,
> >
> > <<Try this one: A chain of 10, 369c generators, octave period.>>
> >
> > Though this sounds more like a loaded mousetrap than a practical
> type
> > question... if small is all your really looking for, how about
> > something on the order of 49/40 and 4375/4096?
> >
> > --Dan Stearns
>
> Actually, with the 10-note 369c MOS, I was looking for a MOS scale
> that Paul would have difficulty finding unison-vectors for, that are
> anything like unisons. i.e. This one was meant to have _big_ UVs, and
> not to contain any good approximations to SWNRs.
>
> Are you asking us to find a linear temperament that treats those
> unison vectors (49/40 and 4375/4096) as commas, and to tell you how
> "good" it is relative to the usual JI criteria.

I think Dan just found unison vectors for your example, Dave!

Graham and Dave, Wilson knew Partch, and his mappings for the Diamond to Modulus-41 and
Modulus-72 keyboards did not use the MIRACLE generator, but rather other generators. So I
don't see how one could say that Partch was using, or implying MIRACLE, in any way
whatsoever.

I wrote,

> Graham and Dave, Wilson knew Partch, and his mappings for the Diamond to Modulus-41
and
> Modulus-72

Oops -- I meant the Partch 43-tone scale, not the diamond.

In-Reply-To: <9h7845+e2fi@eGroups.com>
Paul wrote:

> Graham and Dave, Wilson knew Partch, and his mappings for the Diamond
> to Modulus-41 and Modulus-72 keyboards did not use the MIRACLE
> generator, but rather other generators. So I don't see how one could
> say that Partch was using, or implying MIRACLE, in any way whatsoever.

Oh, come come. If Partch was ever feeling towards Miracle he would have
stopped doing so long before Wilson came up with his Modulus-41 ideas.
That the scale works so well with 41 and 72 does imply Miracle. Then
again, simply using 11-limit JI implies Miracle.

It is interesting that 31, 41 and 72 don't get a mention in Genesis.
Deliberate avoidance of temperaments he can't dismiss so lightly? You
decide!

Graham

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 25, 2001 4:38 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > It was Erv Wilson who hypothesized that Partch was intuitively
> > "feeling out" a version of 41-EDO where two of the pitches could
> > imply either of a pair of ratios (12/11 and 11/10, and their
> > "octave"-complements).
>
> Actually, the pair was 11/10 and 10/9 . . . you don't get a
> PB or CS the other way.

OK, I understand that *theoretically* this is the elegant comparison.

But we had a discussion about this around two years ago...

Didn't Daniel Wolf present cases in Partch's actual compositions
where either pair could be interchangeable? That's what I remember.

(I should have mentioned it the first time around... my bad.)

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Progress seems to have halted on the paper that was to introduce
> MIRACLE . . .
/.../
> If we can do the following math problem, we'll be fine:
>
> Given a k-by-k matrix, containing k-1 commatic unison vectors and 1
> chromatic unison vector, delimiting a periodicity block, find:
>
> (a) the generator of the resulting WF (MOS) scale;
>
> (b) the integer N such that the interval of repetition is 1/N
> octaves.

Can somebody fill me in on what is meant by "interval of
repetition" here?

-C.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > > 2. Masses of people over centuries have effectively given us a
> > > short
> > > > list of those they found useful.
...
> I am [objecting to the above sentence].

I mean the ancient scales that are still in popular use today in
various cultures. eg. "meantone" diatonic. Arabic scales. Various
pentatonics. Gamelan scales.

> > I'll adress the
> > second. Graham Breed (and George Secor) have shown that MIRACLE_41
is
> > almost identical to several of Partch's scales.
>
> Eh . . . not quite.

Err Paul, "almost" is a synonym for "not quite". See my post to the
tuning list entitled "Partch's scales on the Miracle keyboard".

> > I can't help seeing
> > Partch's various scales as gropings towards either Canasta
>
> Don't see it.

No. I was wrong there.

> > or
> > MIRACLE-41.
>
> Toward modulus-41, yes . . . with many other generators functioning
as well as, if not better than,
> the 4/41 (MIRACLE) generator.

No. I'm talking about Miracle-41 and the 7/72 oct generator. 4/41 oct
is only borderline Miracle.

> Yes, Dave, we both want to "rule out" the MOSs with no
approximations to any JI
> intervals/chords (if such a thing is possible). That is where we
(the originators of "MIRACLE")
> differ from Dan Stearns (at least in the viewpoint that goes behind
this paper we're
> contemplating). But that still leaves a great number of
possibilities, as Robert Valentine, for
> example, has been finding.

Oh sure. I was assuming you had read Dan's post and my response to it,
and were referring to that. Sorry.

-- Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> He stopped at 43 in order to make a melodically fairly even scale.
With 10/9 and 11/10 seen as
> a commatic pair (the unison vector involved is 100:99), and their
octave complements another
> such pair, Partch's scale is a 41-tone periodicity block -- or what
Wilson calls a "Constant
> Structure".

I think George Secor, Graham Breed and Dave Keenan disagree with this
analysis, preferring one based on filling in the the diamond gaps
using rationalised Miracle generators. See
/tuning/topicId_25575.html#25575

Does anyone know if Partch regularly used any of the many approximate
JI intervals in his scale such as those with only a 224:225 or 384:385
error (less than 8 cents)?

-- Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Actually, with the 10-note 369c MOS, I was looking for a MOS scale
> > that Paul would have difficulty finding unison-vectors for, that
are
> > anything like unisons. i.e. This one was meant to have _big_ UVs,
and
> > not to contain any good approximations to SWNRs.
> >
> > Are you asking us to find a linear temperament that treats those
> > unison vectors (49/40 and 4375/4096) as commas, and to tell you
how
> > "good" it is relative to the usual JI criteria.
>
> I think Dan just found unison vectors for your example, Dave!

If that's the case, then it makes my point quite well. Isn't it just a
little ridiculous to refer to intervals of 351c and 114c as "unison"
vectors or "commas"?

-- Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Graham and Dave, Wilson knew Partch, and his mappings for the
[43-tone scale] to Modulus-41 and
> Modulus-72 keyboards did not use the MIRACLE generator, but rather
other generators.

Which ones?

> So I
> don't see how one could say that Partch was using, or implying
MIRACLE, in any way
> whatsoever.

All that means is that Partch wasn't intentionally using Miracle and
that Wilson missed the fact that Partch's scales imply it.

-- Dave Keenan

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > > It was Erv Wilson who hypothesized that Partch was intuitively
> > > "feeling out" a version of 41-EDO where two of the pitches could
> > > imply either of a pair of ratios (12/11 and 11/10, and their
> > > "octave"-complements).
> >
> > Actually, the pair was 11/10 and 10/9 . . . you don't get a
> > PB or CS the other way.
>
>
> OK, I understand that *theoretically* this is the elegant
comparison.
>
> But we had a discussion about this around two years ago...
>
> Didn't Daniel Wolf present cases in Partch's actual compositions
> where either pair could be interchangeable? That's what I remember.

It's interesting that Miracle distinguishes all three of these ratios,
as Partch did.

11:12 is -9 generators
10:11 is 22 generators
9:10 is -19 generators

-- Dave Keenan

--- In tuning-math@y..., carl@l... wrote:
> Can somebody fill me in on what is meant by "interval of
> repetition" here?

It's just Paul inventing yet another term for what has been called
(ill advisedly when relating to MOS)
formal octave
interval of equivalence
and more sensibly called
period
interval of periodicity

It gets a little ridiculous referring to 1/29 octave as a formal
octave or an interval of equivalence, as in Graham's 15-limit
temperament.

-- Dave Keenan

I'm replying here to two of Graham's posts about Partch and MIRACLE.

> ----- Original Message -----
> From: <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 25, 2001 2:53 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> One question is, how much did Partch know about Miracle when he drew up
> that original, unpublished scale? It may be stretching credulity to
> suggest he worked it all out, and then pretended it was pure JI. But the
> criteria he was using may well have matched those that are enshrined in
> Miracle. Roughly equal melodic steps will of course favour an MOS. And
> he would have been able to hear the intervals that were almost just by
> Miracle approximations. And so he could have chosen the extra notes to
> maximise these consonances.
>
> In which case, why did he change his mind later? I think it was to get
> more modulation by fifths in the 5-limit plane. With experience, he
> decided this was more important than matching the consonances.
>
> The limitations on modulation by fifths is one of the problems with
> Miracle, at least in a traditional context. Boomsliter and Creel's
> theories work very well with schismic, but not at all well with Miracle,
> temperament.
>

> ----- Original Message -----
> From: <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Cc: <gbreed@cix.compulink.co.uk>
> Sent: Monday, June 25, 2001 7:03 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> Oh, come come. If Partch was ever feeling towards Miracle he would have
> stopped doing so long before Wilson came up with his Modulus-41 ideas.
> That the scale works so well with 41 and 72 does imply Miracle. Then
> again, simply using 11-limit JI implies Miracle.
>
> It is interesting that 31, 41 and 72 don't get a mention in Genesis.
> Deliberate avoidance of temperaments he can't dismiss so lightly? You
> decide!

Graham, you know that I also love speculation!
I'm very impressed by yours here.

John Chalmers is the subscriber on this list who can really
document the relationship between Secor and Partch. (Perhaps
we should also post a query on another list for Kraig Grady?)

I do know, however, that their meeting ocurred quite late in
Partch's life. Partch lamented that Secor's Scalatron was the
instrument he had always wanted, but it came along too late to
do him any good. This was probably early 1970s, possibly late 1960s.

_Genesis_ was published in 1947 or 1949 [1] (1st ed.) and
1974 (2nd ed.), and the only substantial changes in the 2nd edition
concerned Partch's new instruments. The theoretical and historical
sections of the book remained virtually intact.

So I'm certain beyond any doubt that Partch was not *consciously*
aware of MIRACLE before the late 1960s. (note my emphasis)

But Graham's speculations are intriguing, and I'm fairly convinced
by them that Partch *intuitively* understood the MIRACLE concept
and perhaps was indeed guided in constructing his 43-tone scale
by some of the additional "senses" in which the 14 new (and
original 29) pitches could be taken in MIRACLE.

Daniel Wolf, who has had the opportunity to study Partch's
scores in *much* greater depth than I have, has remarked on how
Partch did not always construct his harmonies according to the
lowest-odd-integer hexadic theory presented in _Genesis_.
So perhaps some of these "nonstandard" usages *do* conform
to MIRACLE-like approximations.

Partch's 14 additional pitches are, as Graham correctly states,
primarily an expansion of the Tonality Diamond in the prime-factor-3
dimension, which Graham notes is *not* a feature of MIRACLE.

I've noted before how I thought it was a paradox that for all
his vitriolic abrogation of Pythagoreanism, Partch took exactly
this route in expanding his pitch gamut. It seems that he valued
*something* about traditional music-theory after all, and that
"something" is, again as Graham points out, modulation or
root-movement by 3:2s.

About the equal temperaments discussed in _Genesis_:

First of all, I should say that I was simply writing from memory
before. Now I have the book in front of me, and there are indeed
some ETs that I left out. I'll correct that omission abundantly
now.

Partch (1974, p 417) does make this interesting general observation:

> Fundamentally, equal temperaments are based upon and deduced
> from Pythagorean "cycles," in whole or part.

He opens his chapter on equal-temperaments with a long and
scathing diatribe against 12-EDO, which, by this point in the
book, should not surprise the reader.

Then he discusses the 'First Result of Expansion - "Quartertones"'.
Upon mentioning Carillo, Partch also thus mentions 48- and 96-EDO.

But he actually does go into a little detail about 24-EDO, and
he's even generous enough about its potential to say that
'As a temporary expedient, as an immediately feasible method
of creating new musical resources, "quartertones" are valuable'.
He mentions Haba [which should be spelled H�ba], Hans Barth, and
Mildred Couper and their use of dual regular keyboards, and Meyer
and Moellendorf and their new keyboards.

Then Partch breifly discusses Busoni and 36-EDO, which he characterizes
as "another Polypythagoreanism in tempered expression".

In the middle of this text, on p 430, is Partch's comparative table
of tunings. I will come back to say more about this table after
describing the rest of the text.

Next comes the discussion of Yasser's 19-EDO, then finally 53-EDO.

About Yasser's proposal, Partch emphasizes that its goal is
not the betterment of intonation, but simply an expansion of
scalar resources. He notes the improved approximations to
5- and 7-limit ratios, and also that "The ratios of 7 are somewhat
better also, but still with a maximum falsity of 21.4 cents
(33.1 cents in twelve-tone temperament). The ratios of 11 are
not represented at all". Actually, 19-EDO's closest approximations
to the 11-limit ratios are all between +/- 17.1 and 31.5 cents,
significantly better than 12-EDO's.

Partch had mentioned in "Chapter 15: A Thumbnail Sketch of the
History of Intonation" that King Fang (in China) and Mersenne,
Kircher, and Mercator (in Europe) all proposed this tuning.
In the middle of the discussion of 53-EDO is a digression
"On the Matter of Hearing a 2-Cents Falsity".

Partch notes that 53-EDO is indeed extremely close to 3- and
5-limit JI, but does not consider it suitable for his own use
as it offers little improvement in approximating the 7- and
11-limit ratios he wanted to use.

Finally he examines the keyboard proposals of Nicolaus Ramarinus
(1640) [2], Bosanquet (no date given by Partch, c. 1875?), and
Jas. Paul White (1883) [3].

And that wraps up Partch's "Chapter 17: Equal Temperaments".

Now, back to that comparative table...

Partch's table on p 430 compares his Monophonic 43-tone scale with,
in order:

- 12-EDO,
- 12-tone Pythagorean: a 3^(-6...+5) system,
- 16-tone Meantone: a cycle of implied "5ths" 3^(-5...10) tuned in
1/4-comma meantone, the pair of notes at either end of the cycle
being the additional notes on Handel's organ (according to Partch),
- 17-tone Arabic: a Pythagorean 3^(-12...+4) system,
- 19-EDO,
- 24-EDO,
- 31-EDO,
- 36-EDO,
- 53-EDO.

First, I should note that there are obviously tunings here (the
second, third, and fourth) which are not ETs. Partch had already
discussed these in his "Chapter 16: Polypythagoreanism".

But - SURPRISE! - there's 31-EDO in the table, but
WITH NO MENTION WHATSOEVER IN THE TEXT!!

And I checked all the other chapters in _Genesis_... there's no
mention at all of Huyghens, Fokker, or anything else concerning
31-EDO.

Now THAT'S interesting! ... And I never noticed it before,
having been duped by 31-EDO's appearance in that table into
thinking that Partch said something about it somewhere.

So Graham is right that, except for this inconspicuous little
tabulation, Partch does not mention 31-, 41- or 72-EDO.
Good detective work, Graham!!!

NOTES

[1] I asked before (on the main list) about the actual publication
date. I don't remember now what the outcome was, but I've seen it
listed in catalogs under both dates. The original Preface
is dated April 1947, but the copyright date is 1949.

[2] About Ramarinus, Partch says:
> the "tone" (9/8) was divided into nine "commas",
> according to Hawkins [_History of the Science and
> Practice of Music, vol 1, p 396]. The fifty-third part
> of 2/1 is approximately the width of the "comma" of
> Didymus, 81/80 (21.5 cents; see table above), and since
> six 9/8's are larger than a 2/1 by approximately this
> interval (the "comma" of Pythagoras, 23.5 cents), this
> procedure would result in a fifty-three-tone scale.

Of course, we are well aware that the 9-commas-per-tone
temperament works out to exactly 55-EDO, which is a meantone,
whereas 53-EDO is quasi-just. This choice probably reflects
Partch's own bias; I'd bet that Ramarinus most likely meant
something more like 55-EDO.

[3] Paul (or anyone else in Boston): It still says in the 1974
edition of _Genesis_ that White's harmonium was housed in a
practice room at New England Conservatory, and that Partch
examined it in 1943. I've found page references in _Genesis_
that should have been renumbered from the 1st edition and weren't,
so perhaps this is a story that also should have been updated.
Please... go take a look and let us know!

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

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

> The unison vectors I used for 31+41n are:
>
> [[ 2 -2 2 0 -1]
> [-7 -1 1 1 1]
> [-1 5 0 0 -2]
> [-5 2 2 -1 0]]
>
Getting rid of the first column:

[-2 2 0 -1]
[-1 1 1 1]
[ 5 0 0 -2]
[ 2 2 -1 0]

the resulting FPB is

cents numerator denominator
38.906 45 44
70.672 25 24
79.965 288 275
111.73 16 15
150.64 12 11
182.4 10 9
203.91 9 8
235.68 55 48
262.37 64 55
294.13 32 27
315.64 6 5
347.41 11 9
386.31 5 4
425.22 225 176
427.37 32 25
466.28 72 55
498.04 4 3
536.95 15 11
551.32 11 8
590.22 45 32
609.78 64 45
648.68 16 11
663.05 22 15
701.96 3 2
733.72 55 36
772.63 25 16
774.78 352 225
813.69 8 5
852.59 18 11
884.36 5 3
905.87 27 16
937.63 55 32
964.32 96 55
996.09 16 9
1017.6 9 5
1049.4 11 6
1088.3 15 8
1120 275 144
1129.3 48 25
1161.1 88 45
1200 1 1

>
> That uses 100:99 as the chromatic UV. The more obvious choice
would be a
> schisma, so that
>
> [[-15 8 1 0 0]
> [-7 -1 1 1 1]
> [-1 5 0 0 -2]
> [-5 2 2 -1 0]]
>
> would give the same results.

Again getting rid of the first column, this is

[ 8 1 0 0]
[-1 1 1 1]
[ 5 0 0 -2]
[ 2 2 -1 0]

giving the FPB

cents numerator denominator
31.767 55 54
60.412 729 704
92.179 135 128
111.73 16 15
143.5 88 81
172.14 243 220
203.91 9 8
235.68 55 48
262.37 64 55
296.09 1215 1024
315.64 6 5
347.41 11 9
386.31 5 4
407.82 81 64
439.59 165 128
466.28 72 55
498.04 4 3
519.55 27 20
558.46 243 176
590.22 45 32
609.78 64 45
643.5 1485 1024
670.19 81 55
701.96 3 2
733.72 55 36
760.41 256 165
794.13 405 256
813.69 8 5
845.45 44 27
884.36 5 3
905.87 27 16
937.63 55 32
964.32 96 55
996.09 16 9
1017.6 9 5
1056.5 81 44
1088.3 15 8
1107.8 256 135
1141.5 495 256
1168.2 108 55
1200 2 1

The difference between these two scales is

numerator denominator
242 243
2187 2200
4125 4096
1 1
242 243
2187 2200
1 1
1 1
1 1
32805 32768
1 1
1 1
1 1
99 100
4125 4096
1 1
1 1
99 100
243 242
1 1
1 1
16335 16384
243 242
1 1
1 1
4096 4125
91125 90112
1 1
242 243
1 1
1 1
1 1
1 1
1 1
1 1
243 242
1 1
4096 4125
4125 4096
243 242
2 1

So if the schisma (32805:32768) is the _chromatic_ unison vector of
one of these scales, the two scales are _not_ equivalent, even up to
arbitrary transpositions by _commatic_ unison vectors.

> I can't check this now, as I don't have
> Numerical Python installed, or even Excel. But you may be able
to. Try
> inverting this matrix, and multiplying it by its determinant:
>
[[ 1 0 0 0 0]
[-15 8 1 0 0]
[-7 -1 1 1 1]
[-1 5 0 0 -2]
[-5 2 2 -1 0]]

The determinant is -41, and the inverse is

[ 1 0 0 0 0 ]
[ 65/41 6/41 -2/41 -1/41 -2/41]
[ 95/41 -7/41 16/41 8/41 16/41]
[ 115/41 -2/41 28/41 14/41 -13/41]
[ 142/41 15/41 -5/41 -23/41 -5/41]

> The left hand two columns should be
>
> [[ 41 0]
> [ 65 -6]
> [ 95 7]
> [115 2]
> [142 -15]]

Up to a minus sign, yes.
>
> If they are, the two sets of unison vectors give exactly the same
> results.

They don't!

> I think they must be, because I remember checking the
> determinant before, and any chroma that gives a determinant of 41
when
> placed with Miracle commas should give this result.

Something must be wrong with one of your assumptions.

> You most certainly do need octave-specific matrices. Otherwise,
that
> left-hand column won't be there.

I see that as a good thing . . . don't you?

> There may be an algorithm that works with octave
> invariant matrices, but it's easier to upgrade them to be
> octave-specific, and use a common or garden inverse.

?

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9h7845+e2fi@e...>
> Paul wrote:
>
> > Graham and Dave, Wilson knew Partch, and his mappings for the
Diamond
> > to Modulus-41 and Modulus-72 keyboards did not use the MIRACLE
> > generator, but rather other generators. So I don't see how one
could
> > say that Partch was using, or implying MIRACLE, in any way
whatsoever.
>
> Oh, come come. If Partch was ever feeling towards Miracle he would
have
> stopped doing so long before Wilson came up with his Modulus-41
ideas.

???

> That the scale works so well with 41 and 72 does imply Miracle.

Now you're stretching the meaning of the word "imply".

> Then
> again, simply using 11-limit JI implies Miracle.

Now you're _really_ stretching the meaning of the word "imply"!!! :)

> It is interesting that 31, 41 and 72 don't get a mention in
Genesis.
> Deliberate avoidance of temperaments he can't dismiss so lightly?
You
> decide!

I think he was simply ignorant of these temperaments, in the
literature he was familiar with (which concentrated on 19, 24, and
53). Actually, 31 _is_ in his ET comparison table, isn't it?

Monz wrote:

> _Genesis_ was published in 1947 or 1949 [1] (1st ed.) and
> 1974 (2nd ed.), and the only substantial changes in the 2nd edition
> concerned Partch's new instruments. The theoretical and historical
> sections of the book remained virtually intact.

So, if "Exposition on Monophony" was1933, that's well in advance.

> But Graham's speculations are intriguing, and I'm fairly convinced
> by them that Partch *intuitively* understood the MIRACLE concept
> and perhaps was indeed guided in constructing his 43-tone scale
> by some of the additional "senses" in which the 14 new (and
> original 29) pitches could be taken in MIRACLE.

Be careful you don't get carried away with these speculations. It seems
plausible that he was feeling for something like 41-equal but with improved
11-limit harmony. In that case, you'd expect the result to look something like
a 41-note MOS of a good 11-limit temperament. The scale he ends up with does
fit schismic better than Miracle.

As mathematicians, we should be aware of the dangers of imposing patterns on
data. For the rest, I think the discussion should be taken to the main list if
you think you have a case. Dave Keenan has already come up with some new
arguments.

> Partch's 14 additional pitches are, as Graham correctly states,
> primarily an expansion of the Tonality Diamond in the prime-factor-3
> dimension, which Graham notes is *not* a feature of MIRACLE.
>
> I've noted before how I thought it was a paradox that for all
> his vitriolic abrogation of Pythagoreanism, Partch took exactly
> this route in expanding his pitch gamut. It seems that he valued
> *something* about traditional music-theory after all, and that
> "something" is, again as Graham points out, modulation or
> root-movement by 3:2s.

D'alessandro also ends up with a long chain of 3:2s, and so doesn't work so
well as Miracle.

> And I checked all the other chapters in _Genesis_... there's no
> mention at all of Huyghens, Fokker, or anything else concerning
> 31-EDO.

I thought Fokker did his music theory during the Nazi occupation, hence after
the original publication of Genesis. And Huygens' music theory wouldn't have
been known until then either.

Yasser still suggested eventual evolution to 31 though.

> Now THAT'S interesting! ... And I never noticed it before,
> having been duped by 31-EDO's appearance in that table into
> thinking that Partch said something about it somewhere.
>
> So Graham is right that, except for this inconspicuous little
> tabulation, Partch does not mention 31-, 41- or 72-EDO.
> Good detective work, Graham!!!

With you're detective work we can now say that he avoided *all* consistent
11-limit temperaments!

Graham

"I toss therefore I am" -- Sartre

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > > > 2. Masses of people over centuries have effectively given
us a
> > > > short
> > > > > list of those they found useful.
> ...
> > I am [objecting to the above sentence].
>
> I mean the ancient scales that are still in popular use today in
> various cultures. eg. "meantone" diatonic. Arabic scales. Various
> pentatonics. Gamelan scales.

There are a lot of cultural accidents that lead to "popular use". And
those Gamelan scales . . . you'd need some large unison vectors for
those, wouldn't you?
>
> > > I can't help seeing
> > > Partch's various scales as gropings towards either Canasta
> >
> > Don't see it.
>
> No. I was wrong there.
>
> > > or
> > > MIRACLE-41.
> >
> > Toward modulus-41, yes . . . with many other generators
functioning
> as well as, if not better than,
> > the 4/41 (MIRACLE) generator.
>
> No. I'm talking about Miracle-41 and the 7/72 oct generator. 4/41
oct
> is only borderline Miracle.

I meant 4/41 in a modulus-41, not 41-tET, sense. Doesn't the 19/72
generator work as well for Partch's scale as the 7/72 generator?

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > He stopped at 43 in order to make a melodically fairly even
scale.
> With 10/9 and 11/10 seen as
> > a commatic pair (the unison vector involved is 100:99), and their
> octave complements another
> > such pair, Partch's scale is a 41-tone periodicity block -- or
what
> Wilson calls a "Constant
> > Structure".
>
> I think George Secor, Graham Breed and Dave Keenan disagree with
this
> analysis, preferring one based on filling in the the diamond gaps
> using rationalised Miracle generators. See
> /tuning/topicId_25575.html#25575

The analyses are not necessarily incompatible!!!

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > Actually, with the 10-note 369c MOS, I was looking for a MOS
scale
> > > that Paul would have difficulty finding unison-vectors for,
that
> are
> > > anything like unisons. i.e. This one was meant to have _big_
UVs,
> and
> > > not to contain any good approximations to SWNRs.
> > >
> > > Are you asking us to find a linear temperament that treats
those
> > > unison vectors (49/40 and 4375/4096) as commas, and to tell you
> how
> > > "good" it is relative to the usual JI criteria.
> >
> > I think Dan just found unison vectors for your example, Dave!
>
> If that's the case, then it makes my point quite well. Isn't it
just a
> little ridiculous to refer to intervals of 351c and 114c
as "unison"
> vectors or "commas"?
>
Only if you think of the scale as existing _initially_ in JI. Some of
my favorite scales, such as the 14-out-of-26-tET 7-limit scales,
involve very large unison vectors. Since they are tempered out over a
large number of consonant intervals, the fact that they are very
large in JI doesn't bother me.

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> All that means is that Partch wasn't intentionally using Miracle

You bet!

> and
> that Wilson missed the fact that Partch's scales imply it.

"Imply" is a little strong. I'd say "suggest".

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> It's interesting that Miracle distinguishes all three of these
ratios,
> as Partch did.
>
> 11:12 is -9 generators
> 10:11 is 22 generators
> 9:10 is -19 generators
>
> -- Dave Keenan

Partch distinguished them because they're all in the diamond. Early
on, his scale was _just_ the diamond. 10:11 and 9:10 (and their
octave complements) are the _only_ pair of notes in the diamond that
fall in the same place in modulus-41 . . . that's why Partch ended up
with 43, rather than 41, notes . . . he was not willing to compromise
the diamond . . . he built many instruments around it!

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> But Graham's speculations are intriguing, and I'm fairly convinced
> by them that Partch *intuitively* understood the MIRACLE concept
> and perhaps was indeed guided in constructing his 43-tone scale
> by some of the additional "senses" in which the 14 new (and
> original 29) pitches could be taken in MIRACLE.

I will continue to take the (partly devil's advocate) stance that
this is not the case at all and Partch was really just feeling out
modulus-41 while steadfastly maintaining the diamond intact.
>
> Partch's 14 additional pitches are, as Graham correctly states,
> primarily an expansion of the Tonality Diamond in the prime-factor-3
> dimension, which Graham notes is *not* a feature of MIRACLE.

Good evidence for my position -- note how well 41-tET approximates
prime-factor-3.

> About Yasser's proposal, Partch emphasizes that its goal is
> not the betterment of intonation, but simply an expansion of
> scalar resources. He notes the improved approximations to
> 5- and 7-limit ratios, and also that "The ratios of 7 are somewhat
> better also, but still with a maximum falsity of 21.4 cents
> (33.1 cents in twelve-tone temperament). The ratios of 11 are
> not represented at all". Actually, 19-EDO's closest approximations
> to the 11-limit ratios are all between +/- 17.1 and 31.5 cents,
> significantly better than 12-EDO's.

Perhaps Partch *intuitively* understood that 19-tET was not
consistent in the 11-limit :)

>
> [3] Paul (or anyone else in Boston): It still says in the 1974
> edition of _Genesis_ that White's harmonium was housed in a
> practice room at New England Conservatory, and that Partch
> examined it in 1943. I've found page references in _Genesis_
> that should have been renumbered from the 1st edition and weren't,
> so perhaps this is a story that also should have been updated.
> Please... go take a look and let us know!
>

Hmm . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > > > > 2. Masses of people over centuries have effectively given
> us a
> > > > > short
> > > > > > list of those they found useful.
> > ...
> > > I am [objecting to the above sentence].
> >
> > I mean the ancient scales that are still in popular use today in
> > various cultures. eg. "meantone" diatonic. Arabic scales. Various
> > pentatonics. Gamelan scales.
>
> There are a lot of cultural accidents that lead to "popular use".
And
> those Gamelan scales . . . you'd need some large unison vectors for
> those, wouldn't you?

Yeah. Dammit. :-) So neither PBs nor JI seem relevant there, except
possibly in a Setharian sense.

> Doesn't the 19/72
> generator work as well for Partch's scale as the 7/72 generator?

That's a JI minor third, so kleismic, generator. You tell me. How many
holes in a chain that encompasses it? How big are the errors?. Maybe
on the main list.
-- Dave Keenan

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> Be careful you don't get carried away with these speculations. It
seems
> plausible that he was feeling for something like 41-equal but with
improved
> 11-limit harmony.

Oh yes, that's certainly still worth considering.

> In that case, you'd expect the result to look
something like
> a 41-note MOS of a good 11-limit temperament. The scale he ends up
with does
> fit schismic better than Miracle.

Please give details. How many holes in a chain that encompasses it.
How big are the errors? Are there any overloads? Maybe on the other
list.
-- Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> I will continue to take the (partly devil's advocate) stance that
> this is not the case at all

Yes. Please do.

> and Partch was really just feeling out
> modulus-41 while steadfastly maintaining the diamond intact.

But Partch did compromise the diamond in the 39 note "Ur" scale, and
in just such a way as to reduce its width on a Miracle chain, i.e.
deleting the 11/10 and 20/11.

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > I will continue to take the (partly devil's advocate) stance that
> > this is not the case at all
>
> Yes. Please do.
>
> > and Partch was really just feeling out
> > modulus-41 while steadfastly maintaining the diamond intact.
>
> But Partch did compromise the diamond in the 39 note "Ur" scale,
and
> in just such a way as to reduce its width on a Miracle chain, i.e.
> deleting the 11/10 and 20/11.
>
I replied to this view on the tuning list.

> ----- Original Message -----
> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 25, 2001 10:25 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > > > It was Erv Wilson who hypothesized that Partch was intuitively
> > > > "feeling out" a version of 41-EDO where two of the pitches could
> > > > imply either of a pair of ratios (12/11 and 11/10, and their
> > > > "octave"-complements).
> > >
> > > Actually, the pair was 11/10 and 10/9 . . . you don't get a
> > > PB or CS the other way.
> >
> > OK, I understand that *theoretically* this is the elegant
> > comparison.
> >
> > But we had a discussion about this around two years ago...
> >
> > Didn't Daniel Wolf present cases in Partch's actual compositions
> > where either pair could be interchangeable? That's what I remember.
>
> It's interesting that Miracle distinguishes all three of these ratios,
> as Partch did.
>
> 11:12 is -9 generators
> 10:11 is 22 generators
> 9:10 is -19 generators

Yes, Dave, exactly! This is another reason why I tend to agree
with Graham's speculations (and yours?) that Partch was intuitively
"feeling out" MIRACLE even moreso than 41-EDO.

(BTW, I made a webpage out of that post I sent earlier today
... unfortunately, didn't finish it before I had to go to work.
Coming soon, to a web-browser near you...)

-monz
http://www.monz.org
"All roads lead to n^0"

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> ----- Original Message -----
> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, June 25, 2000 12:58 PM
> Subject: Re: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> ... Actually, 19-EDO's closest approximations
> to the 11-limit ratios are all between +/- 17.1 and 31.5 cents,
> significantly better than 12-EDO's.
>
> Partch had mentioned in "Chapter 15: A Thumbnail Sketch of the
> History of Intonation" that King Fang (in China) and Mersenne,
> Kircher, and Mercator (in Europe) all proposed this tuning.

Sorry... I had shifted some text around and left this bit unedited
by mistake. That last clause should read "... all proposed 53-EDO".

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning-math/message/329

> Sorry about the previous message, sent by mistake.
>
> --- In tuning-math@y..., jpehrson@r... wrote:
> > I'm getting a little confused here... Did Harry Partch use a
> > 41-tone scale in addition to his 43-tone scale??
>
> Sure. There are two in the Scala archive, But that's not what I
meant.
>
> > He never actually used "Miracle 41" did he??
>
> No. I didn't say that either. But he might not have noticed if
someone
> had substituted a scale which was MIRACLE-41 plus a couple of extra
> notes from MIRACLE-45.
>
> Read:
> http://www.anaphoria.com/secor.PDF
> and
> http://x31eq.com/decimal_lattice.htm#partch
> and then tell me what you don't understand.
>
> Regards,
> -- Dave Keenan

Hi Dave...

I guess what I'm understanding is that some of the "fill in" notes
that Partch used to complete his 43-tone scale could be described by
the "miracle generator..."

Am I on the right track??

___________ __________ _______
Joseph Pehrson

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 25, 2001 3:15 PM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > But Graham's speculations are intriguing, and I'm fairly convinced
> > by them that Partch *intuitively* understood the MIRACLE concept
> > and perhaps was indeed guided in constructing his 43-tone scale
> > by some of the additional "senses" in which the 14 new (and
> > original 29) pitches could be taken in MIRACLE.
>
> I will continue to take the (partly devil's advocate) stance that
> this is not the case at all and Partch was really just feeling out
> modulus-41 while steadfastly maintaining the diamond intact.

OK, Paul... I can see your point of view as well.

But I find it *more* than very interesting that Partch
knew about 31-EDO's good approximations to a significant
percentage of his scale, and chose to say *nothing* about it!

> Perhaps Partch *intuitively* understood that 19-tET was not
> consistent in the 11-limit :)

Yes, that's a very good suggestion.

And in an earlier post, Paul wrote:

> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > All that means is that Partch wasn't intentionally using Miracle
> > and that Wilson missed the fact that Partch's scales imply it.
>
>
> "Imply" is a little strong. I'd say "suggest".

Hmmm... I think you have a very good point there, Paul.
"Suggest" is more likely what I mean too, when I said "imply".

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

/tuning-math/message/366

>
> So Graham is right that, except for this inconspicuous little
> tabulation, Partch does not mention 31-, 41- or 72-EDO.
> Good detective work, Graham!!!
>

So the thought is that, possibly, something was "bothering" him about
these temperaments... (??)

_________ _________ ________
Joseph Pehrson

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
>
> But I find it *more* than very interesting that Partch
> knew about 31-EDO's good approximations to a significant
> percentage of his scale, and chose to say *nothing* about it!

That's understandable, since 31-tET conflates pairs of ratios in his diamond, such as 9:8 and
10:9, and gives them both an error of 11 cents! Since these were primary consonances in
Partch's system, and 11 cent errors were almost unthinkably large to Partch, the dismissal is not
surprising. Plus, you might say, he was utterly predisposed to dismissing any ET on the
principle of the thing.

--- In tuning-math@y..., jpehrson@r... wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> /tuning-math/message/366
>
> >
> > So Graham is right that, except for this inconspicuous little
> > tabulation, Partch does not mention 31-, 41- or 72-EDO.
> > Good detective work, Graham!!!
> >
>
> So the thought is that, possibly, something was "bothering" him about
> these temperaments... (??)
>
That's kind of silly. He did include 31 in his table, and was unfamiliar with 41 and 72, both absent
from the literature with which he was familiar. But yes, he was predisposed toward dismissing
any ET, and probably wasn't in a hurry to go about finding a "good" one.

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 25, 2001 9:13 PM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> >
> > But I find it *more* than very interesting that Partch
> > knew about 31-EDO's good approximations to a significant
> > percentage of his scale, and chose to say *nothing* about it!
>
> That's understandable, since 31-tET conflates pairs of ratios
> in his diamond, such as 9:8 and 10:9, and gives them both an
> error of 11 cents! Since these were primary consonances in
> Partch's system, and 11 cent errors were almost unthinkably
> large to Partch, the dismissal is not surprising.

Paul, thanks so much for your insight into this. More below.

> Plus, you might say, he was utterly predisposed to dismissing any
> ET on the principle of the thing.

Yes, I would have said something like this myself.

I think "diamondic" is indeed the paradigm which best characterizes
Partch's feelings about his scale.

This whole thread about a possible MIRACLE intuition guiding
Partch has made it abundantly clear to me that the literal
structures embedded in the Tonality Diamond were of paramount
importance to him.

Since arguably the thing the Diamond shows best is the
at-least-dual nature of each ratio, which is a property
Partch emphasized repeatedly was inherent in ratios (quite
obvious to my mind, since they're a relationship described
by two numbers, duh!), then it seems to me to follow that
this dual property was perhaps the primary conceptual focus
of his tuning system.

If this is the case, then I find that to be a very valuable
insight into Partch's _modus operandi_.

It's also fascinating that Partch was more interested in
expanding his harmonic resources along Pythagorean lines
(pun intended) rather than the higher-prime relationships
approximated by MIRACLE.

I'm interested now more than ever in knowing some of Daniel
Wolf's knowledge and opinions on this subject. A full-scale
analysis of the *non*-JI harmonies in Partch's compositions
would reveal a ton of information.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
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--- In tuning-math@y..., jpehrson@r... wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > No. I didn't say that either. But he might not have noticed if
> someone
> > had substituted a scale which was MIRACLE-41 plus a couple of
extra
> > notes from MIRACLE-45.

> I guess what I'm understanding is that some of the "fill in" notes
> that Partch used to complete his 43-tone scale could be described by
> the "miracle generator..."
>
> Am I on the right track??

Yes. But I was referring to the early 43 toner in "Expositions on
Monophony". It only applies to the later one in "Genesis" if Partch
wouldn't have noticed you'd switched to his earlier scale.

-- Dave Keenan

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
Paul Erlich wrote:
> > That's understandable, since 31-tET conflates pairs of ratios
> > in his diamond.

This is useful terminology.
"<temperament> conflates ratios in <JI structure>"
means the same as
"<JI structure> overloads <temperament>".

> This whole thread about a possible MIRACLE intuition guiding
> Partch has made it abundantly clear to me that the literal
> structures embedded in the Tonality Diamond were of paramount
> importance to him.
>
> Since arguably the thing the Diamond shows best is the
> at-least-dual nature of each ratio, which is a property
> Partch emphasized repeatedly was inherent in ratios (quite
> obvious to my mind, since they're a relationship described
> by two numbers, duh!), then it seems to me to follow that
> this dual property was perhaps the primary conceptual focus
> of his tuning system.
>
> If this is the case, then I find that to be a very valuable
> insight into Partch's _modus operandi_.
>
>
> It's also fascinating that Partch was more interested in
> expanding his harmonic resources along Pythagorean lines
> (pun intended) rather than the higher-prime relationships
> approximated by MIRACLE.

What do you mean here by "higher-prime". I hope you only mean 5, 7 and
11.

But it seems that he went Miracle at first and then later changed only
four notes for Pythagorean. He changed only 49/48 to 81/80 and 27/20
to 15/11 (and their inversions).

So I can postulate 3 forces in historical order: First Diamondic, then
Miracle (which simply means that he wanted to fill in the diamond gaps
while minimising the number of extra notes and maximising the number
of 11-limit consonances, both strict and with small errors) and
finally the old Pythagorean/Diatonic reasserted itself sightly.

> I'm interested now more than ever in knowing some of Daniel
> Wolf's knowledge and opinions on this subject. A full-scale
> analysis of the *non*-JI harmonies in Partch's compositions
> would reveal a ton of information.

Yes indeed. We might be able to better answer the "schismic vs.
miracle" question based on that.

-- Dave Keenan

In-Reply-To: <9h8hfa+ki4n@eGroups.com>
Dave Keenan wrote:

> > In that case, you'd expect the result to look
> something like
> > a 41-note MOS of a good 11-limit temperament. The scale he ends up
> with does
> > fit schismic better than Miracle.
>
> Please give details. How many holes in a chain that encompasses it.
> How big are the errors? Are there any overloads? Maybe on the other
> list.

The 43 notes become a 41 note schismic MOS, with duplicates exactly where
you expect them. Wilson showed this. You get the same 41 note MOS with
either the Exposition of Monophony or Genesis 43 note scales.

Graham

> ----- Original Message -----
> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, June 26, 2001 12:08 AM
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> This is useful terminology.
> "<temperament> conflates ratios in <JI structure>"
> means the same as
> "<JI structure> overloads <temperament>".

I agree.
Any of you want to write a couple of good definitions for me?

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > It's also fascinating that Partch was more interested in
> > expanding his harmonic resources along Pythagorean lines
> > (pun intended) rather than the higher-prime relationships
> > approximated by MIRACLE.
>
> What do you mean here by "higher-prime". I hope you only
> mean 5, 7 and 11.

Good catch, Dave... I should have been more clear about that
myself. Yes, that's exactly what I mean. I was differentiating
between "traditional" Pythagorean root-movement and the
possibilities offered collectively by 5, 7, and 11.

>
> But it seems that he went Miracle at first and then later changed only
> four notes for Pythagorean. He changed only 49/48 to 81/80 and 27/20
> to 15/11 (and their inversions).
>
> So I can postulate 3 forces in historical order: First Diamondic, then
> Miracle (which simply means that he wanted to fill in the diamond gaps
> while minimising the number of extra notes and maximising the number
> of 11-limit consonances, both strict and with small errors) and
> finally the old Pythagorean/Diatonic reasserted itself sightly.

Hmmm... at this point, I think I really should dig out my copy of
Richard Kassel's dissertation "The Evolution of Partch's Monophony".
It explains in detail all the early and intermediate stages in his
theory, including tabulations of all his different scales.

-monz
http://www.monz.org
"All roads lead to n^0"

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Paul wrote:

> > If they are, the two sets of unison vectors give exactly the same
> > results.
>
> They don't!
>
> > I think they must be, because I remember checking the
> > determinant before, and any chroma that gives a determinant of 41
> when
> > placed with Miracle commas should give this result.
>
> Something must be wrong with one of your assumptions.

Yes, they both give Miracle41, but a different Miracle41 each time/

> > You most certainly do need octave-specific matrices. Otherwise,
> that
> > left-hand column won't be there.
>
> I see that as a good thing . . . don't you?

No, it helps to define the temperament.

If you invert and normalize the octave-invariant matrix, the left hand column
gives you the prime intervals in terms of generators. If there's a common
factor, divide through by it, and call it the octave division.

The only problems are those anomalous cases where the determinant is a multiple
of the temperament you want. So octave-specific are still winning.

> > There may be an algorithm that works with octave
> > invariant matrices, but it's easier to upgrade them to be
> > octave-specific, and use a common or garden inverse.
>
> ?

Okay, inverting the octave invariant matrices still tells you something. So
how do we spot the anomalies?

I'll update to <http://x31eq.com/vectors.html> if I remember. The
unison vector finder is slightly improved in that it finds something for the
multiple-29 scale now.

Graham

"I toss therefore I am" -- Sartre

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > This is useful terminology.
> > "<temperament> conflates ratios in <JI structure>"
> > means the same as
> > "<JI structure> overloads <temperament>".
>
> I agree.
> Any of you want to write a couple of good definitions for me?

Two or more notes (ratios) of the JI structure become a single note of
the temperament. For example 9/8 and 10/9 are replaced by a single "D"
in meantone temperaments. So we say that meantone conflates 9/8 with
10/9 or that any JI structure containing _both_ 9/8 and 10/9 overloads
meantone.

> > But it seems that he went Miracle at first and then later changed
only
> > four notes for Pythagorean. He changed only 49/48 to 81/80 and
27/20
> > to 15/11 (and their inversions).

Sorry. That should have been "changed ... 15/11 to 27/20". I typed
that pair back to front.

> > So I can postulate 3 forces in historical order: First Diamondic,
then
> > Miracle (which simply means that he wanted to fill in the diamond
gaps
> > while minimising the number of extra notes and maximising the
number
> > of 11-limit consonances, both strict and with small errors) and
> > finally the old Pythagorean/Diatonic reasserted itself sightly.
>
> Hmmm... at this point, I think I really should dig out my copy of
> Richard Kassel's dissertation "The Evolution of Partch's Monophony".
> It explains in detail all the early and intermediate stages in his
> theory, including tabulations of all his different scales.

Sounds great.
-- Dave Keenan

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

/tuning-math/message/404
>
> Since arguably the thing the Diamond shows best is the
> at-least-dual nature of each ratio, which is a property
> Partch emphasized repeatedly was inherent in ratios (quite
> obvious to my mind, since they're a relationship described
> by two numbers, duh!), then it seems to me to follow that
> this dual property was perhaps the primary conceptual focus
> of his tuning system.
>

A question:

In arithmetic and mathematics is the *numerator* of a fraction ever
considered "more important" than the *denominator?*

Or is that a silly question...? It seems to me in simple arithmetic,
the numerator seems more "impressive..." maybe because the numbers
are larger??

Just as in "otonal??" Hasn't the "otonal" series, on the overall,
been considered *significantly* more important than the *utonal* over
the years??

Or am I just "out to lunch..."

_________ _______ ______
Joseph Pehrson

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning-math/message/407

> > I'm interested now more than ever in knowing some of Daniel
> > Wolf's knowledge and opinions on this subject. A full-scale
> > analysis of the *non*-JI harmonies in Partch's compositions
> > would reveal a ton of information.
>
> Yes indeed. We might be able to better answer the "schismic vs.
> miracle" question based on that.
>
> -- Dave Keenan

Doesn't this imply that, somehow, Partch was using the "non-JI"
harmonies in a different way than his "JI" harmonies??

Personally, I would doubt that. Once he had his scale, he probably
just used it "as is" regardless of the derivation of the notes..

??

__________ ________ ________
Joseph Pehrson

On Wed, 27 Jun 2001 jpehrson@rcn.com wrote:

> A question:
>
> In arithmetic and mathematics is the *numerator* of a fraction ever
> considered "more important" than the *denominator?*

Not that I know of -- see the definition of the rational numbers as equivalence
classes of ordered pairs of integers. In an ordered pair, *somebody's* gotta
be number one and somebody else's gotta be number two :-). Ya ain't got no
ordered pair otherwise :-).

> Or is that a silly question...? It seems to me in simple arithmetic,
> the numerator seems more "impressive..." maybe because the numbers
> are larger??

> Just as in "otonal??" Hasn't the "otonal" series, on the overall,
> been considered *significantly* more important than the *utonal* over
> the years??

Outside of Partch, yes -- Otonal/Major is *musically* more important than
Utonal/Minor *in common practice Western music*. One of the things Partch was
trying to do, after having defined Otonal and Utonal to begin with, was to
treat them equally in his music and right what he considered to be a wrong in
this respect. I haven't heard enough of his music to know whether Otonal and
Utonal are in fact equally respected in his works.

> Or am I just "out to lunch..."

Are you buying? :-)
--
znmeb@aracnet.com (M. Edward Borasky) http://www.aracnet.com/~znmeb

How to Stop A Folksinger Cold # 2
"Are you going to Scarborough Fair?..."
No.

--- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote:

/tuning-math/message/425

> On Wed, 27 Jun 2001 jpehrson@r... wrote:
>
> > A question:
> >
> > In arithmetic and mathematics is the *numerator* of a fraction
ever considered "more important" than the *denominator?*
>
> Not that I know of -- see the definition of the rational numbers as
equivalence classes of ordered pairs of integers. In an ordered pair,
*somebody's* gotta be number one and somebody else's gotta be number
two :-). Ya ain't got no ordered pair otherwise :-).
>

Got it! Thanks, Ed!

>
> > Just as in "otonal??" Hasn't the "otonal" series, on the overall,
> > been considered *significantly* more important than the *utonal*
over the years??
>
> Outside of Partch, yes -- Otonal/Major is *musically* more
important than Utonal/Minor *in common practice Western music*. One
of the things Partch was trying to do, after having defined Otonal
and Utonal to begin with, was to treat them equally in his music and
right what he considered to be a wrong in this respect. I haven't
heard enough of his music to know whether Otonal and Utonal are in
fact equally respected in his works.

Gee... this is an interesting question, but Jon Szanto isn't on this
list... Maybe I'll post something to the "biggie..."

>
> > Or am I just "out to lunch..."
>
> Are you buying? :-)

Sure! But, unfortunately... you're in Oregon at the moment.... :)

_________ _______ _____
Joseph Pehrson

--- In tuning-math@y..., jpehrson@r... wrote:
> In arithmetic and mathematics is the *numerator* of a fraction ever
> considered "more important" than the *denominator?*

No. I don't think so. It's all completely dual.

> Or is that a silly question...?

No. Its a good question.

> It seems to me in simple
arithmetic,
> the numerator seems more "impressive..." maybe because the numbers
> are larger??

In ordinary (non-musical) usage the numerator is just as likely to be
smaller than the denominator.

> Just as in "otonal??" Hasn't the "otonal" series, on the overall,
> been considered *significantly* more important than the *utonal*
over
> the years??

Yes. But this doesn't make the numerator or denominator special. It
makes _the_smallest_of_the_two_numbers_ special. Several frequencies
having their fundamentals ocurring as if they are the harmonics of a
lower virtual fundamental, gives more consonance than several
frequencies that each have one harmonic corresponding to a higher
"guide-tone".

In the case of octave-equivalent pitches we have a convention to put
them in a form that is between 1/1 and 2/1 so they have positive
logarithms. But for non octave-equivalent pitches we can have 2/3
different from 3/2.

For intervals, octave equivalence doesn't matter. 2:3 describes
exactly the same interval as 3:2. I have argued before for a
convention of putting the small number first, as we do for "extended
ratios" such as 4:5:6. But when we want to take its logarithm (to
convert to cents) we will still enter it as 3/2, i.e. big number as
numerator, so that we are dealing with positive logarithms.

But remember these are only conventions or conveniences. The musical
specialness is in "big number versus little number", not "numerator
versus denominator".

Regards,
-- Dave Keenan

--- In tuning-math@y..., Graham
Breed <graham@m...> wrote:

> Yes, they both give Miracle41, but a different Miracle41 each time/

Can you explain what you mean
by "different"? They're both
centered around the 1/1, so it's
not the mode that's different . . .
>
> If you invert and normalize the octave-invariant matrix, the left hand column
> gives you the prime intervals in terms of generators.

Well that sounds like it solves the
Hypothesis in a demonstrative
fashion, yes?

--- In tuning-math@y...,
jpehrson@r... wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> /tuning-math/message/407
>
> > > I'm interested now more than ever in knowing some of Daniel
> > > Wolf's knowledge and opinions on this subject. A full-scale
> > > analysis of the *non*-JI harmonies in Partch's compositions
> > > would reveal a ton of information.
> >
> > Yes indeed. We might be able to better answer the "schismic vs.
> > miracle" question based on that.
> >
> > -- Dave Keenan
>
> Doesn't this imply that, somehow, Partch was using the "non-JI"
> harmonies in a different way than his "JI" harmonies??

Well a question can't imply a fact.
But if you mean, doesn't it
_assume_ that, then no. In fact,
the more Partch used them in the
same way, the easier it will be to
decide which unison vectors he
may have accepted.
>
> Personally, I would doubt that. Once he had his scale, he probably
> just used it "as is" regardless of the derivation of the notes..
>
> ??
>
Unfortunately, that may be a bit
too much to hope for. Partch
devised an involved
compositional apparatus in
_Genesis_ based on JI harmonies,
and I would be shocked if this
didn't still guide his later works
somewhat.

> --- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote:

> >
> > Outside of Partch, yes -- Otonal/Major is *musically* more
> important than Utonal/Minor *in common practice Western music*.

On what basis do you make that
claim? They seem to be equal
enough in importance in this
music to "fool" Riemann, Partch,
and many other theorists to give
them equal footing a priori.

I was paraphrasing Partch ... I can probably find the line in _Genesis_, but
one of his goals was to restore Untonality to equal footing with Otonality,
thus implying an existing *in*equality. I believe the trigger for this was
Hindemith referring to the minor as a "clouding" of the major. I'm not
familiar enough with the bulk of Hindemith's work to know if his *music*
reflects this or not; what I've *heard* of Hindemith I rather like, being a
flute player :-). If it really matters I can dig it up. I was certainly not
implying that *Partch* treated them as unequal!

--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

> -----Original Message-----
> From: Paul Erlich [mailto:paul@stretch-music.com]
> Sent: Wednesday, June 27, 2001 6:38 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: 41 "miracle" and 43 tone scales
>
>
> > --- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote:
>
> > >
> > > Outside of Partch, yes -- Otonal/Major is *musically* more
> > important than Utonal/Minor *in common practice Western music*.
>
> On what basis do you make that
> claim? They seem to be equal
> enough in importance in this
> music to "fool" Riemann, Partch,
> and many other theorists to give
> them equal footing a priori.
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

--- In tuning-math@y..., "M.
Edward Borasky" <znmeb@a...>
wrote:
> I was paraphrasing Partch ... I can probably find the line in _Genesis_, but
> one of his goals was to restore Untonality to equal footing with Otonality,
> thus implying an existing *in*equality.

In musical _theories_ -- not in
any of the musical _practice_ that
he liked, as he understood it.

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning-math/message/428
>
> But remember these are only conventions or conveniences. The
musical specialness is in "big number versus little number",
not "numerator versus denominator".
>
> Regards,
> -- Dave Keenan

Got it! Thanks, Dave!

________ _______ ______
Joseph Pehrson

In-Reply-To: <9he0uv+d2ll@eGroups.com>
Paul wrote:

> > Yes, they both give Miracle41, but a different Miracle41 each time/
>
> Can you explain what you mean
> by "different"? They're both
> centered around the 1/1, so it's
> not the mode that's different . . .

One is 10+41n, the other 31+41n. The mapping by period and generator is
the same both times. So they're both aspects of the same temperament.
It depends on whether you take this "set of MOS scales" result seriously.
It doesn't come out of the octave invariant method discussed below.

> > If you invert and normalize the octave-invariant matrix, the left
> > hand column
> > gives you the prime intervals in terms of generators.
>
> Well that sounds like it solves the
> Hypothesis in a demonstrative
> fashion, yes?

If you can prove it will always work. I can't, but am pleased it does.
You can certainly always define the scale in terms of some kind of
octave-invariant interval, and call that the generator. Perhaps that's
all it comes down to. But I've always said this was obvious from the
matrix technique. But showing that the unison vectors lead to a linear
temperament is different from showing they give a CS periodicity block,
or whatever it is you asked.

The octave-specific method doesn't always give a result. It fails with
the unison vectors I'm using for the multiple-29 temperament. But you
can always define a temperament in terms of a pair of intervals, even if
they aren't the ones you want for the MOS.

The octave-invariant result for multiple-29, BTW, is this mapping:

[0, 707281, 707281, 707281, 707281]

when I wanted

[0, 29, 29, 29, 29]

Incidentally, an alternative octave-specific case would be to define an
extra chromatic unison vector instead of the octave. The the two left
hand columns of the inverse will be the mapping by scale steps.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9he0uv+d2ll@e...>
> Paul wrote:
>
> > > Yes, they both give Miracle41, but a different Miracle41 each
time/
> >
> > Can you explain what you mean
> > by "different"? They're both
> > centered around the 1/1, so it's
> > not the mode that's different . . .
>
> One is 10+41n, the other 31+41n.

What do you mean by this notation?

> The mapping by period and generator is
> the same both times. So they're both aspects of the same
temperament.
> It depends on whether you take this "set of MOS scales" result
seriously.

I'm not following you.

> It doesn't come out of the octave invariant method discussed below.

What's "It" in this sentence?

>
> > > If you invert and normalize the octave-invariant matrix, the
left
> > > hand column
> > > gives you the prime intervals in terms of generators.
> >
> > Well that sounds like it solves the
> > Hypothesis in a demonstrative
> > fashion, yes?
>
> If you can prove it will always work. I can't, but am pleased it
does.
> You can certainly always define the scale in terms of some kind of
> octave-invariant interval, and call that the generator. Perhaps
that's
> all it comes down to.

Yes, but this choice should be unique . . . there should only be one
(octave-invariant) generator.

> But I've always said this was obvious from the
> matrix technique. But showing that the unison vectors lead to a
linear
> temperament is different from showing they give a CS periodicity
block,
> or whatever it is you asked.

Well there may be some differences in our understanding of this, as
the above (different miracle-41s) may be indicating. But I think
we're on the right track . . . ?
>
> The octave-specific method doesn't always give a result.

Uh-oh. So maybe I can convince you to switch over to octave-invariant?

> It fails with
> the unison vectors I'm using for the multiple-29 temperament. But
you
> can always define a temperament in terms of a pair of intervals,
even if
> they aren't the ones you want for the MOS.

Don't they _have_ to be the generator and the interval of repetition?
>
> The octave-invariant result for multiple-29, BTW, is this mapping:
>
> [0, 707281, 707281, 707281, 707281]
>
> when I wanted
>
> [0, 29, 29, 29, 29]

Can you explain how the number 707281 comes about?

> > One is 10+41n, the other 31+41n.
>
> What do you mean by this notation?

Temperements including the ETs with 10+41n or 31+41n notes, where n is a
non-negative integer.

> > The mapping by period and generator is
> > the same both times. So they're both aspects of the same
> temperament.
> > It depends on whether you take this "set of MOS scales" result
> seriously.
>
> I'm not following you.

I explained this before. When you generate the scales from a set of unison
vectors, one of them chromatic, the natural result is something like 10+41n
rather than a single MOS or the full range of temperaments defined by the
commatic unison vectors.

> > It doesn't come out of the octave invariant method discussed below.
>
> What's "It" in this sentence?

The restricted set of temperaments. But in fact I was wrong there. In fact,
the second column of the normalized octave-specific inverse is the same as the
first column of the octave-invariant one, but with an extra zero. I didn't
notice it was the generator mapping before, but managed to get the right
results anyway :)

> > If you can prove it will always work. I can't, but am pleased it
> does.
> > You can certainly always define the scale in terms of some kind of
> > octave-invariant interval, and call that the generator. Perhaps
> that's
> > all it comes down to.
>
> Yes, but this choice should be unique . . . there should only be one
> (octave-invariant) generator.

This brings us back to

""""
The determinant is -41, and the inverse is
[ 1 0 0 0 0 ]
[ 65/41 6/41 -2/41 -1/41 -2/41]
[ 95/41 -7/41 16/41 8/41 16/41]
[ 115/41 -2/41 28/41 14/41 -13/41]
[ 142/41 15/41 -5/41 -23/41 -5/41]

> The left hand two columns should be
>
> [[ 41 0]
> [ 65 -6]
> [ 95 7]
> [115 2]
> [142 -15]]

Up to a minus sign, yes.
>
> If they are, the two sets of unison vectors give exactly the same
> results.

They don't!
"""

There are aways two generators that will work. The minus sign differentiates
them.

> > But I've always said this was obvious from the
> > matrix technique. But showing that the unison vectors lead to a
> linear
> > temperament is different from showing they give a CS periodicity
> block,
> > or whatever it is you asked.
>
> Well there may be some differences in our understanding of this, as
> the above (different miracle-41s) may be indicating. But I think
> we're on the right track . . . ?

Oh, unquestionably.

> > The octave-specific method doesn't always give a result.
>
> Uh-oh. So maybe I can convince you to switch over to octave-invariant?

I think it would be worth writing a script that only uses them. It would mean
altering the code in temper.py to accept a mapping by generators, so it's a bit
of work.

> > It fails with
> > the unison vectors I'm using for the multiple-29 temperament. But
> you
> > can always define a temperament in terms of a pair of intervals,
> even if
> > they aren't the ones you want for the MOS.
>
> Don't they _have_ to be the generator and the interval of repetition?

No. If you take this matrix at face value:

> [[ 41 0]
> [ 65 -6]
> [ 95 7]
> [115 2]
> [142 -15]]/41

it defines Miracle using one 41st part of an octave, and a 41st part of the
usual generator. That works, but it isn't efficient.

> > The octave-invariant result for multiple-29, BTW, is this mapping:
> >
> > [0, 707281, 707281, 707281, 707281]
> >
> > when I wanted
> >
> > [0, 29, 29, 29, 29]
>
> Can you explain how the number 707281 comes about?

It's 29^4. I'm sure it means I chose the chromatic unison vector wrongly. The
interesting thing is that the generator matrix is a multiple of what it should
be. In fact, the whole matrix has a common factor, which may be the clue that
something's wrong. Although dividing through by that common factor won't
work. Also, this is a case where the inverse of the octave-specific matrix
doesn't get the generator mapping right.

If the method almost works with an arbitrary chroma, that means we're a step
towards getting it to work with only commatic unison vectors, which should be
possible.

Graham

"I toss therefore I am" -- Sartre

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> > > One is 10+41n, the other 31+41n.
> >
> > What do you mean by this notation?
>
> Temperements including the ETs with 10+41n or 31+41n notes, where n
is a
> non-negative integer.

I'm still confused about how there can be two different MIRACLE-41s.
Are there two different Canastas too, or does the divergence only
happen at 41?
>
> > > The mapping by period and generator is
> > > the same both times. So they're both aspects of the same
> > temperament.
> > > It depends on whether you take this "set of MOS scales" result
> > seriously.
> >
> > I'm not following you.
>
> I explained this before. When you generate the scales from a set
of unison
> vectors, one of them chromatic, the natural result is something
like 10+41n
> rather than a single MOS or the full range of temperaments defined
by the
> commatic unison vectors.

A single MOS is what I expect. The number of notes in that MOS
normally equals the determinant of the matrix of unison vectors,
including the chromatic one. So where are we disagreeing?

> >
> > Yes, but this choice should be unique . . . there should only be
one
> > (octave-invariant) generator.
>
> This brings us back to
>
> """"
> The determinant is -41, and the inverse is
> [ 1 0 0 0 0 ]
> [ 65/41 6/41 -2/41 -1/41 -2/41]
> [ 95/41 -7/41 16/41 8/41 16/41]
> [ 115/41 -2/41 28/41 14/41 -13/41]
> [ 142/41 15/41 -5/41 -23/41 -5/41]
>
> > The left hand two columns should be
> >
> > [[ 41 0]
> > [ 65 -6]
> > [ 95 7]
> > [115 2]
> > [142 -15]]
>
> Up to a minus sign, yes.
> >
> > If they are, the two sets of unison vectors give exactly the same
> > results.
>
> They don't!
> """
>
> There are aways two generators that will work. The minus sign
differentiates
> them.

But if you center the resulting scale around 1/1, either the plus-
sign or the minus-sign generator should give the same results. So
that can't account for the difference we saw.
> >
> > Don't they _have_ to be the generator and the interval of
repetition?
>
> No. If you take this matrix at face value:
>
> > [[ 41 0]
> > [ 65 -6]
> > [ 95 7]
> > [115 2]
> > [142 -15]]/41
>
> it defines Miracle using one 41st part of an octave, and a 41st
part of the
> usual generator. That works, but it isn't efficient.

How does it work? Certainly the scale doesn't repeat itself every
41st of an octave.

> Also, this [the 29th-of-an-octave thing] is a case where the
inverse of the octave-specific matrix
> doesn't get the generator mapping right.

:(
>
> If the method almost works with an arbitrary chroma, that means
we're a step
> towards getting it to work with only commatic unison vectors, which
should be
> possible.

Well you _should_ be able to find the generator without specifying
the chroma, but you need the chroma to select a particular MOS.

Paul Erlich wrote:

> I'm still confused about how there can be two different MIRACLE-41s.
> Are there two different Canastas too, or does the divergence only
> happen at 41?

There are two Canstas, 10+31n and 21+31n.

> A single MOS is what I expect. The number of notes in that MOS
> normally equals the determinant of the matrix of unison vectors,
> including the chromatic one. So where are we disagreeing?

It's not clear to me if the duality is real or not.

> > There are aways two generators that will work. The minus sign
> differentiates
> > them.
>
> But if you center the resulting scale around 1/1, either the plus-
> sign or the minus-sign generator should give the same results. So
> that can't account for the difference we saw.

Are the FPBs different in this sense? For the matrices, it's because the
mapping to steps in the MOS is always the same.

> > No. If you take this matrix at face value:
> >
> > > [[ 41 0]
> > > [ 65 -6]
> > > [ 95 7]
> > > [115 2]
> > > [142 -15]]/41
> >
> > it defines Miracle using one 41st part of an octave, and a 41st
> part of the
> > usual generator. That works, but it isn't efficient.
>
> How does it work? Certainly the scale doesn't repeat itself every
> 41st of an octave.

Yes, it would do. If you try tuning a 12-note meantone in cents relative
to 12-equal, you'll see the pattern.

> > If the method almost works with an arbitrary chroma, that means
> we're a step
> > towards getting it to work with only commatic unison vectors, which
> should be
> > possible.
>
> Well you _should_ be able to find the generator without specifying
> the chroma, but you need the chroma to select a particular MOS.

Indeed so! But the octave invariant matrix doesn't give you that
particular MOS. Although it gives you enough of a clue to work it out
from the determinant, the main result is the mapping in terms of
generators.

Graham

--- In tuning-math@y..., graham@m... wrote:
> Paul Erlich wrote:
>
> > I'm still confused about how there can be two different MIRACLE-
41s.
> > Are there two different Canastas too, or does the divergence only
> > happen at 41?
>
> There are two Canstas, 10+31n and 21+31n.

Hmmm . . . what's the _real_ difference between these two?
>
> Are the FPBs different in this sense?

Yes -- look back a few days -- I showed that there was a schisma
difference between a few corresponding pitches in the two FPBs, even
though you're claiming the schisma as a chromatic unison vector
(hence one that isn't tempered out).

> > How does it work? Certainly the scale doesn't repeat itself every
> > 41st of an octave.
>
> Yes, it would do. If you try tuning a 12-note meantone in cents
relative
> to 12-equal, you'll see the pattern.

I see the pattern, but that doesn't make 1/12 octave the period of a
12-note meantone . . . ?

> > Well you _should_ be able to find the generator without
specifying
> > the chroma, but you need the chroma to select a particular MOS.
>
> Indeed so! But the octave invariant matrix doesn't give you that
> particular MOS.

Sure it does! Just take the determinant (usually)! (Assuming you
already know the generator.)

> Although it gives you enough of a clue to work it out
> from the determinant,

A big clue!

> the main result is the mapping in terms of
> generators.

Well, that does seem to be something very interesting you've found.
How can we get that without plugging in a chroma at all?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> > > > One is 10+41n, the other 31+41n.
> > >
> > > What do you mean by this notation?
> >
> > Temperements including the ETs with 10+41n or 31+41n notes, where
n
> is a
> > non-negative integer.

Graham,

I still don't understand this. So 10+41n includes ETs 10 51 61 71 ...
and 31+41n includes 31 72 113 ... Only the second looks anything like
Miracle to me.

Paul Erlich wrote:
> I'm still confused about how there can be two different MIRACLE-41s.
> Are there two different Canastas too, or does the divergence only
> happen at 41?

Graham replied:
> There are two Canstas, 10+31n and 21+31n.

What could this mean when 31 and 72 aren't members of either of these
series? I'm very confused.

-- Dave Keenan

In-Reply-To: <9hgb48+f78j@eGroups.com>
Dave Keenan wrote:

> I still don't understand this. So 10+41n includes ETs 10 51 61 71 ...
> and 31+41n includes 31 72 113 ... Only the second looks anything like
> Miracle to me.

They both cover this part of the scale tree

31 10

41

72 51

93 113 91 61

but branch differently at 41. So one is more closely associated with the
Miracle family, but I don't think there's anything special about one
unison vector as compared to the other.

Probably I should ignore the generalisation, and take 31+10 and 10+31 as
different ways of writing the same MOS.

> Graham replied:
> > There are two Canstas, 10+31n and 21+31n.
>
> What could this mean when 31 and 72 aren't members of either of these
> series? I'm very confused.

31 is the member where n=infinity. It enshrines the relationship
10+21=31, and hence this part of the scale tree

21 10

31

52 41
73 83 72 51

There are two different ways you can move on from Canasta, and different
chromas suggest different branchings, but I've yet to see a deep reason
for it.

Graham

In-Reply-To: <9hg94q+1es0@eGroups.com>
Paul wrote:

> > There are two Canstas, 10+31n and 21+31n.
>
> Hmmm . . . what's the _real_ difference between these two?

How are you defining reality?

> > Are the FPBs different in this sense?
>
> Yes -- look back a few days -- I showed that there was a schisma
> difference between a few corresponding pitches in the two FPBs, even
> though you're claiming the schisma as a chromatic unison vector
> (hence one that isn't tempered out).

So does that mean the schisma isn't a valid chroma?

> > Yes, it would do. If you try tuning a 12-note meantone in cents
> relative
> > to 12-equal, you'll see the pattern.
>
> I see the pattern, but that doesn't make 1/12 octave the period of a
> 12-note meantone . . . ?

It makes it the period of a linear temperament that includes 12-note
meantone as a subset.

> > > Well you _should_ be able to find the generator without
> specifying
> > > the chroma, but you need the chroma to select a particular MOS.
> >
> > Indeed so! But the octave invariant matrix doesn't give you that
> > particular MOS.
>
> Sure it does! Just take the determinant (usually)! (Assuming you
> already know the generator.)

Usually isn't good enough, we're looking for proof here. Besides,
taking the determinant's cheating. It doesn't mean anything for
octave-invariant matrices, but happens to be part of the result for
octave-specific matrices.

> > the main result is the mapping in terms of
> > generators.
>
> Well, that does seem to be something very interesting you've found.
> How can we get that without plugging in a chroma at all?

I'm hoping that always using a fifth for the top row will work. If not,
framing the problem might help. We want to find a generator consistent
with the simplest mapping, I suppose. Which means minimizing the
determinant. We don't want it to go to zero, but that follows from the
matrix being invertible.

Is there any established theory of integer matrices, or discrete vector
spaces, we can latch on to?

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9hg94q+1es0@e...>
> Paul wrote:
>
> > > There are two Canstas, 10+31n and 21+31n.
> >
> > Hmmm . . . what's the _real_ difference between these two?
>
> How are you defining reality?

Tuning.
>
> > > Are the FPBs different in this sense?
> >
> > Yes -- look back a few days -- I showed that there was a schisma
> > difference between a few corresponding pitches in the two FPBs,
even
> > though you're claiming the schisma as a chromatic unison vector
> > (hence one that isn't tempered out).
>
> So does that mean the schisma isn't a valid chroma?

I'm not saying that . . . but first, can you determine which of the
two scales (if either) is the "real" MIRACLE-41 (to within commatic
unison vectors)?
>
> > > Yes, it would do. If you try tuning a 12-note meantone in
cents
> > relative
> > > to 12-equal, you'll see the pattern.
> >
> > I see the pattern, but that doesn't make 1/12 octave the period
of a
> > 12-note meantone . . . ?
>
> It makes it the period of a linear temperament that includes 12-
note
> meantone as a subset.

Oh -- but a very strange subset. Any "normal" subset should repeat
exactly at the period . . . or that's how I've been thinking about
this stuff.
>
> > > > Well you _should_ be able to find the generator without
> > specifying
> > > > the chroma, but you need the chroma to select a particular
MOS.
> > >
> > > Indeed so! But the octave invariant matrix doesn't give you
that
> > > particular MOS.
> >
> > Sure it does! Just take the determinant (usually)! (Assuming you
> > already know the generator.)
>
> Usually isn't good enough, we're looking for proof here. Besides,
> taking the determinant's cheating. It doesn't mean anything for
> octave-invariant matrices,

It doesn't mean anything?? It means a lot -- see the "Gentle
Introduction" again . . .

> but happens to be part of the result for
> octave-specific matrices.
>
Part of the __________ result?

> > > the main result is the mapping in terms of
> > > generators.
> >
> > Well, that does seem to be something very interesting you've
found.
> > How can we get that without plugging in a chroma at all?
>
> I'm hoping that always using a fifth for the top row will work. If
not,
> framing the problem might help. We want to find a generator
consistent
> with the simplest mapping, I suppose.

The simplest mapping? Not following you. The generator of an MOS is
unique.

Paul wrote:

> > > > There are two Canstas, 10+31n and 21+31n.
> > >
> > > Hmmm . . . what's the _real_ difference between these two?
> >
> > How are you defining reality?
>
> Tuning.

There's no difference.

> > So does that mean the schisma isn't a valid chroma?
>
> I'm not saying that . . . but first, can you determine which of the
> two scales (if either) is the "real" MIRACLE-41 (to within commatic
> unison vectors)?

Don't know, it's all on my Linux partition.

> > It makes it the period of a linear temperament that includes 12-
> note
> > meantone as a subset.
>
> Oh -- but a very strange subset. Any "normal" subset should repeat
> exactly at the period . . . or that's how I've been thinking about
> this stuff.

Yes, it takes 144 notes to get the 12 note scale. But it does prove the
hypothesis that every set of vectors gives some linear temperament.

> > Usually isn't good enough, we're looking for proof here. Besides,
> > taking the determinant's cheating. It doesn't mean anything for
> > octave-invariant matrices,
>
> It doesn't mean anything?? It means a lot -- see the "Gentle
> Introduction" again . . .

I'll look it up.

> > but happens to be part of the result for
> > octave-specific matrices.
> >
> Part of the __________ result?

Octave-specific. It's the top left-hand corner.

> > I'm hoping that always using a fifth for the top row will work. If
> not,
> > framing the problem might help. We want to find a generator
> consistent
> > with the simplest mapping, I suppose.
>
> The simplest mapping? Not following you. The generator of an MOS is
> unique.

As long as it's unique, there's no problem.

(Technically, it'll be +/-, but that's all negotiable)

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9hgb48+f78j@e...>
> Dave Keenan wrote:
>
> > I still don't understand this. So 10+41n includes ETs 10 51 61 71
...
> > and 31+41n includes 31 72 113 ... Only the second looks anything
like
> > Miracle to me.
>
> They both cover this part of the scale tree
>
> 31 10
>
>
> 41
>
> 72 51
>
> 93 113 91 61

Ok. I can see that you are the one who is confused here. Miracle does
not go outside of

31

41

72

93 113

Well Ok, it does go a tiny bit past 41, but nowhere near all the way
to 10. Just as 5-EDO is nothing like a meantone.

This is because, outside of 31 to 41(and-a-bit) there are better 7
or 11-limit approximations than the ones used by MIRACLE.

> > Graham replied:
> > > There are two Canstas, 10+31n and 21+31n.
> >
> > What could this mean when 31 and 72 aren't members of either of
these
> > series? I'm very confused.
>
> 31 is the member where n=infinity.

Huh? When n=oo 10+31n and 21+31n also go to oo. I think you must be
talking your own language here.

It enshrines the relationship
> 10+21=31, and hence this part of the scale tree
>
>
> 21 10
>
>
> 31
>
> 52 41
> 73 83 72 51
>
>
> There are two different ways you can move on from Canasta

Maybe so. But only one of them is MIRACLE.

On 6/29/01 7:48 PM, "Dave Keenan" <D.KEENAN@UQ.NET.AU> wrote:

>> 31 10
>>
>>
>> 41
>>
>> 72 51
>>
>> 93 113 91 61
>

Shouldn't the bottom line be 103, 113, 92, 61?

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
> On 6/29/01 7:48 PM, "Dave Keenan" <D.KEENAN@U...> wrote:
>
> >> 31 10
> >>
> >>
> >> 41
> >>
> >> 72 51
> >>
> >> 93 113 91 61
> >
>
> Shouldn't the bottom line be 103, 113, 92, 61?

Oh yes. Well spotted!

On 6/30/01 2:37 AM, "Dave Keenan" <D.KEENAN@UQ.NET.AU> wrote:

>> Shouldn't the bottom line be 103, 113, 92, 61?
>
> Oh yes. Well spotted!

Okay. Just trying to follow along.

Dave Keenan wrote:

> > They both cover this part of the scale tree
> >
> > 31 10
> >
> >
> > 41
> >
> > 72 51
> >
> > 93 113 91 61
>
> Ok. I can see that you are the one who is confused here. Miracle does
> not go outside of
>
> 31
>
>
> 41
>
> 72
>
> 93 113

I see a 41 there.

> Well Ok, it does go a tiny bit past 41, but nowhere near all the way
> to 10. Just as 5-EDO is nothing like a meantone.

So decimal notation is now invalid? And blackjack isn't part of the family?
5-EDO may not be a meantone, but pentatonic scales certainly are.

> This is because, outside of 31 to 41(and-a-bit) there are better 7
> or 11-limit approximations than the ones used by MIRACLE.

> > > Graham replied:
> > > > There are two Canstas, 10+31n and 21+31n.
> > >
> > > What could this mean when 31 and 72 aren't members of either of
> these
> > > series? I'm very confused.
> >
> > 31 is the member where n=infinity.
>
> Huh? When n=oo 10+31n and 21+31n also go to oo. I think you must be
> talking your own language here.

Canasta is made up of 31 steps. For 10+31n, there are 10 of those at
(n+1)/(10+31n) octaves and the other 21 are n/(10+31n) octaves. As n tends
to infinity, both steps tend to 1/31 octaves.

> It enshrines the relationship
> > 10+21=31, and hence this part of the scale tree
> >
> >
> > 21 10
> >
> >
> > 31
> >
> > 52 41
> > 73 83 72 51
> >
> >
> > There are two different ways you can move on from Canasta
>
> Maybe so. But only one of them is MIRACLE.

I've changed the way the temperaments are written to sweep all this under
the carpet.

Graham

"I toss therefore I am" -- Sartre

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:
> So decimal notation is now invalid?

Of course not.

> And blackjack isn't part of the family?

Of course it is part of the family.

> 5-EDO may not be a meantone, but pentatonic scales certainly are.

Indeed, this is the crux of the confusion.

The Stern-Brocot tree (considered as fractions of an octave) knows
nothing about odd-limits (or any other kind), while the definition of
Miracle, or meantone or any other temperament, must refer to them. The
tree can tell us two different things about a temperament.
(a) The number of notes in its MOS
(b) The EDOs that are included in that temperament
But we look up these things on the scale tree in two different ways.

We need to know the range of generator sizes that are within the
temperament. First we determine what limit we are using (say 7-odd for
Miracle, 5-odd for meantone). Then we consider the maximum number of
generators we are willing to chain to approximate these just
intervals. (say 20 for Miracle and 11 for meantone). From this we can
determine the range of generator sizes for which the temperament's
mapping from primes to generators (Miracle [6, -7, -2] and meantone
[1, 4]) gives us the best approximation. It is really the mapping from
primes to generators that is the definition of the temperament.

Once we have the two extreme generator sizes, we express these as
fractions of an octave and mark them at the "bottom" of the tree
(where the reals live). Draw straight lines up from these and the
denominator of any fraction between those bounds gives us an ET within
that temperament. The denominator of any fraction reachable by going
up the tree from these, gives us the cardinality of a MOS in the
temperament. So 10 and 11 and 21 are MOS cardinalities in Miracle
temperament but certainly not EDO cardinalities. If that were the
case, why stop at 10 and 11, why not go all the way back to 0 and 1?

And remember that the SB tree has numerators and denominators. For
convenience when talking about a particular temperament we drop the
numerators. This might lead to confusion if we join together what are
really disjoint parts of the tree, based on the denominators only.

> I've changed the way the temperaments are written to sweep all this
under
> the carpet.

I'm glad.

-- Dave Keenan

I have a question for all of you mathematicians.

I've just put up a Dictionary entry for LucyTuning.
http://www.ixpres.com/interval/dict/lucy.htm

In it, I'd like to provide the calculation for the
ratio of the LucyTuning "5th". Can this be simplified?:

( 2^(3 / (2*PI) ) ) * ( {2 / [2^(5 / (2*PI) ) ] } ^(1/2) )

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

Yup! Shore can be simplified :-)
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

> -----Original Message-----
> From: monz [mailto:joemonz@yahoo.com]
> Sent: Saturday, June 30, 2001 6:40 PM
> To: tuning-math@yahoogroups.com
> Subject: Re: [tuning-math] Re: Hypothesis revisited
>
>
> I have a question for all of you mathematicians.
>
> I've just put up a Dictionary entry for LucyTuning.
> http://www.ixpres.com/interval/dict/lucy.htm
>
>
> In it, I'd like to provide the calculation for the
> ratio of the LucyTuning "5th". Can this be simplified?:
>
> ( 2^(3 / (2*PI) ) ) * ( {2 / [2^(5 / (2*PI) ) ] } ^(1/2) )
>
>
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> I have a question for all of you mathematicians.
>
> I've just put up a Dictionary entry for LucyTuning.
> http://www.ixpres.com/interval/dict/lucy.htm
>
>
> In it, I'd like to provide the calculation for the
> ratio of the LucyTuning "5th". Can this be simplified?:
>
> ( 2^(3 / (2*PI) ) ) * ( {2 / [2^(5 / (2*PI) ) ] } ^(1/2) )

I think so. The LucyTuning "major third" is 2^(1/pi). Add two octaves to form the "major
seventeenth": 2^(2+1/pi). Take the fourth root (since it's a meantone, the fifth will be the fourth
root of the major seventeenth): 2^(1/2 + 1/(4*pi)). Is that right?

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, June 30, 2001 7:00 PM
> Subject: [tuning-math] Re: Hypothesis revisited
>

> I think so. The LucyTuning "major third" is 2^(1/pi).
> Add two octaves to form the "major seventeenth": 2^(2+1/pi).
> Take the fourth root (since it's a meantone, the fifth
> will be the fourth root of the major seventeenth):
> 2^(1/2 + 1/(4*pi)). Is that right?

Thanks for this great explanation, Paul.

Your answer is slightly different from the one Ed Borasky
calculated with Derive:

2^( (2*pi) + 1 / (4*pi) )

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

Both simplify to the same thing.

--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

> -----Original Message-----
> From: Paul Erlich [mailto:paul@stretch-music.com]
> Sent: Saturday, June 30, 2001 7:00 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Hypothesis revisited
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > I have a question for all of you mathematicians.
> >
> > I've just put up a Dictionary entry for LucyTuning.
> > http://www.ixpres.com/interval/dict/lucy.htm
> >
> >
> > In it, I'd like to provide the calculation for the
> > ratio of the LucyTuning "5th". Can this be simplified?:
> >
> > ( 2^(3 / (2*PI) ) ) * ( {2 / [2^(5 / (2*PI) ) ] } ^(1/2) )
>
> I think so. The LucyTuning "major third" is 2^(1/pi). Add two
> octaves to form the "major
> seventeenth": 2^(2+1/pi). Take the fourth root (since it's a
> meantone, the fifth will be the fourth
> root of the major seventeenth): 2^(1/2 + 1/(4*pi)). Is that right?
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: Paul Erlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Saturday, June 30, 2001 7:00 PM
> > Subject: [tuning-math] Re: Hypothesis revisited
> >
>
> > I think so. The LucyTuning "major third" is 2^(1/pi).
> > Add two octaves to form the "major seventeenth": 2^(2+1/pi).
> > Take the fourth root (since it's a meantone, the fifth
> > will be the fourth root of the major seventeenth):
> > 2^(1/2 + 1/(4*pi)). Is that right?
>
>
> Thanks for this great explanation, Paul.
>
> Your answer is slightly different from the one Ed Borasky
> calculated with Derive:
>
> 2^( (2*pi) + 1 / (4*pi) )
>
It's completely different.

2^( (2*pi) + 1 / (4*pi) ) = 82.2967 = 7635.3¢

2^(1/2 + 1/(4*pi)) = 1.4944 = 695.49¢

Dave Keenan wrote 25-6:
>If that's the case, then it makes my point quite well. Isn't it just a
>little ridiculous to refer to intervals of 351c and 114c as "unison"
>vectors or "commas"?

I was thinking that too. Since Fokker used the terms defining and
constructing intervals, we could say "commatic defining interval"
and "chromatic defining interval". (The constructing intervals are
the prime base intervals like 3/2, 5/4, 7/4, etc.)

Manuel