Tuning-Math messages 301 - 325

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Message: 301

Date: Sat, 23 Jun 2001 08:06:45

Subject: Re: Hypothesis revisited

From: 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


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Message: 302

Date: Sat, 23 Jun 2001 14:04:51

Subject: Re: Hypothesis revisited

From: Graham Breed

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


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Message: 303

Date: Sat, 23 Jun 2001 18:27:20

Subject: Re: Hypothesis revisited

From: Graham Breed

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
<Unison vector to MOS script *>.  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


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Message: 305

Date: Fri, 23 Jun 2000 11:02:43

Subject: Re: Hypothesis revisited

From: monz

> From: Graham Breed <graham@m...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> 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
Yahoo! GeoCities *
"All roads lead to n^0"


 




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Message: 306

Date: Sat, 23 Jun 2001 19:23:41

Subject: Re: Hypothesis revisited

From: Graham Breed

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


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Message: 307

Date: Sat, 23 Jun 2001 19:41:08

Subject: Re: Hypothesis revisited

From: Graham Breed

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


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Message: 308

Date: Sun, 24 Jun 2001 03:04:39

Subject: Re: Hypothesis revisited

From: jpehrson@r...

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

Yahoo groups: /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


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Message: 309

Date: Sun, 24 Jun 2001 03:09:34

Subject: Re: Hypothesis revisited

From: jpehrson@r...

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

Yahoo groups: /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


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Message: 310

Date: Sun, 24 Jun 2001 03:11:54

Subject: Re: Hypothesis revisited

From: jpehrson@r...

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

Yahoo groups: /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.


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Message: 311

Date: Sat, 23 Jun 2001 22:34:37

Subject: Re: Hypothesis revisited

From: monz

> ----- Original Message ----- 
> From: Graham Breed <graham@m...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> 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
Yahoo! GeoCities *
"All roads lead to n^0"


 


_________________________________________________________
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Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 313

Date: Sun, 24 Jun 2001 08:36:34

Subject: [celestial-tuning] common musical intervals in Babylonian math texts

From: monz

This post was originally for celestial-tuning,
but it got so math-heavy that I decided to send a
copy to tuning-math.

> From: monz <joemonz@y...>
> To: <celestial-tuning@xxxxxxxxxxx.xxx>
> Sent: Friday, June 22, 2001 5:08 PM
> Subject: Re: [celestial-tuning] Sumerian tuning
>
> I actually did find a paper on the web somewhere which was
> devoted to a discussion of the syntonic comma in ancient
> Babylonian mathematics.  It was more in relation to other
> uses than music (can't remember what right now, mainly
> architecture and building).  Try a search on "comma" and
> "Babylonian", see what you find.

> ----- Original Message -----
> From: <JGill99@i...>
> To: <celestial-tuning@xxxxxxxxxxx.xxx>
> Sent: Saturday, June 23, 2001 2:49 PM
> Subject: [celestial-tuning] Sumerian Links
>

> I found a couple of links searching. One relates to
> the Sumerians and Babylonians in relation to their
> mathematics, with some materials involving their
> music (cross-referenced abstracts of existing articles
> and books):
>
> Sumerians and Babylonians - Mathematics and the Liberal Arts *


Thanks for this link, Jay!  Looks like a lot of good reading
on ancient math.


I found the article I was referring to:
Stereometric Texts *


It's part of _A History of Measures_ by Livio C. Stecchini.

The author's native language appears to be Italian, because
this article (in English) is *full* of typos, so its already
difficult discussion is even harder to follow than it need be.

This article, as I correctly remembered a few days ago,
is about the volumes, weights, and specific gravities
of various amounts of bricks.  As the author says in
section 7:

> Considering the importance of bricks in the life of
> Mesopotamia it is not surprising that the units of
> volume and weight were so well adjusted the problem
> of measuring and transporting bricks.



Here are some extracts with the discussion of the Greek
terms which appear in music-theory.


Near the end of section 2:

> (a cube that contains 300 double qa with a six-finger
> edge, less a discrepancy komma or 81/80),

(A "qa" is a measurement of area.)

Stecchini explicitly equates the "komma" with 81:80.


From section 4:

> A massiqtu of 60 qa has a base of 24 x 24 fingers and
> a height of 22½ (that is, 24 minus a diesis).

(A "finger" is a measurement of length.)

Thus, this "diesis" is 24 : 22&1/2 = 16:15.


In the next paragraph, however:

> The tablet indicates that when one came to the sheqel,
> the unit used to weigh the media of exchange, one
> reckoned by referring to the basic sheqel of 9 grams:
> here the unit is a double sheqel of 18 grams reduced
> of a diesis; it is equal to two sheqels of 8.4 grams.

Thus, this is a different "diesis": 9 : 8.4 = 15:14.


Then in section 6 we find:

> ... we find a relation 1:3 3/5 between the two amounts
> of seed, with a discrepancy diesis (3 3/8 * 16/15 = 3 3/5).

So here again the "diesis" is 16:15.


At the end of section 6:

> The text describes a near-cube with a basis of 4 x 4 fingers
> and a height of 3 3/4. If it were a perfect cube the qa
> would be 64 cubic fingers with an excess of a diesis;

The volume of the given shape is 60 cubic fingers, so this
diesis is again 64:60 = 16:15.


> 7. Neugebauer and Sachs have shown that a brick
> measuring 15 x 10 x 5 fingers was considered a typical
> brick, but they have not explained why such a brick
> should have been considered typical. The explanation
> is that the brick has the volume of a royal qa:
> 750 cubic fingers by the barley cubit equal
> 216 cubic fingers by the great cubit. However,
> the brick is calculated with an excess of a leimma
> over the volume of the qa, for the purpose of obtaining
> a brick measured by be convenient figures 15 x 10 x 5
> barley fingers. Deducting a leimma the volume of a
> brick becomes 720 cubic fingers. Since 720 = 216 x 3 1/3,
> the relation between royal qa and normal qa is calculated
> as 1:3 1/3, a relation frequently used instead of the
> less convenient relation 1:3 3/8, with a resulting
> discrepancy komma. The dimensions of the brick are
> such that 6 occupy the area of a square cubit and
> 36 have the volume of a cubic cubit. Below it will
> appear that these bricks were counted by the dozen,
> and that a dozen of bricks is as much as a man can
> carry. Calculating by the great cubit each brick is
> a royal qa with an excess of a leimma, so that 120 bricks,
> or 10 dozens, make exactly a cube of great cubit.


Note that what Stecchini is calling a "leimma" is *not* the
same as the Pythagorean musical interval of that name (it's
also the Pythagorean "chromatic semitone"), which is the
ratio 256:243, which translates into ~90.2249957 cents.

This "leimma" is 750:720 = 25:24, the familiar 5-limit
"chromatic semitone", which translates into ~70.6724269 cents.

This "discrepancy komma" is again the same ratio
as the musical syntonic comma: 3&3/8 : 3&1/3 = 81:80.

Note that Stecchini also mentions a "diesis" here.  I'm a
little confused about exactly how big this is.  It seems to
describe the difference between the "mina stereometric brutta"
and the "regular mina", and I gather from the discussion that
the number of sheqels contained in these minai are respectively
63&2/3 and 60, which gives the ratio 191:180, ~102.690878 cents.

But I could be wrong here.  The ratio between the two
specific gravities is 2.4 : 2.25 = 16:15.  The earlier
mentions of diesis usually show it to be 16:15.

Actually, my other calculations may also be wrong.
I think a very close study of this article (as well as the
others in the series by Stecchini) is necessary to be sure
about what these terms really mean.




-monz
Yahoo! GeoCities *
"All roads lead to n^0"





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Message: 314

Date: Sun, 24 Jun 2001 08:59:28

Subject: Re: [celestial-tuning] common musical intervals in Babylonian math texts

From: monz

Wow... look what I found in the appendix to Stecchini's
series of articles on ancient measurement:

APPENDIX *


> I have given to the discrepancies names derived from
> the accidentals of musical scales, because there is
> a close correlation between units of measures and
> ancient musical scales.  This is made clear by Chinese
> musical treatises. Actually I have found that the
> reading of the Greek musical treatises or of the
> similar Chinese ones, which must have been derived
> from a common source, is the best preparation for the
> understanding of the arithmetic of ancient metrology.
>
> The arithmetic of discrepancies is essential to the
> understanding of the development of problems in
> cuneiform mathematical texts.  It is disputed among
> musicologists whether musical scales have a physiological
> or conventional origin. The evidence I have gathered
> indicates that musical scales were derived from the
> arrangement of the units of volume. The ancients used
> to arrange the units of public reference standards in
> a series, in ascending or descending order. The relation
> among the contents of the basic units of volume appears
> to have been adopted as determining the basic tetrachord.


Stecchini then gives examples, and quite a bit of further
discussion on music and tuning.



-monz
Yahoo! GeoCities *
"All roads lead to n^0"


 


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Message: 315

Date: Sun, 24 Jun 2001 09:05:48

Subject: Re: Hypothesis revisited

From: monz

> ----- Original Message ----- 
> From: <jpehrson@r...>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, June 23, 2001 8:11 PM
> Subject: [tuning-math] Re: Hypothesis revisited
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> Yahoo groups: /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
Yahoo! GeoCities *
"All roads lead to n^0"


 


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Message: 316

Date: Sun, 24 Jun 2001 11:32:15

Subject: Re: common musical intervals in Babylonian math texts

From: monz

> ----- Original Message -----
> From: monz <joemonz@y...>
> To: <celestial-tuning@xxxxxxxxxxx.xxx>; <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, June 24, 2001 8:59 AM
> Subject: [tuning-math] Re: [celestial-tuning] common musical intervals in
Babylonian math texts
>
>
> Wow... look what I found in the appendix to Stecchini's
> series of articles on ancient measurement:
>
> APPENDIX *
>
>
> > I have given to the discrepancies names derived from
> > the accidentals of musical scales, because there is
> > a close correlation between units of measures and
> > ancient musical scales.


I was just talking with Brink and had this idea:

If the Sumerians based their musical interval measurements
on the same ones they used for building physical brick structures,
they could possibly have viewed a wall (for example) as a
lattice-diagram!

Would some of the geometers out there take a look at
Stecchini's articles
Units of Length *
Stereometric Texts *

and construct some plausible examples?  I'd be very
interested in that, but am having a hard time understanding
the proliferation of different measurement units.

If I can get a better understanding, perhaps I'll give
it a try.



-monz
Yahoo! GeoCities *
"All roads lead to n^0"





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Message: 317

Date: Sun, 24 Jun 2001 21:17:59

Subject: Re: Hypothesis revisited

From: Paul Erlich

--- 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.


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Message: 318

Date: Sun, 24 Jun 2001 21:26:51

Subject: Re: Hypothesis revisited

From: Paul Erlich

--- 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.


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Message: 319

Date: Sun, 24 Jun 2001 21:30:41

Subject: Re: Hypothesis revisited

From: Paul Erlich

--- 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.


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Message: 320

Date: Sun, 24 Jun 2001 23:58:22

Subject: Re: Hypothesis revisited

From: Dave Keenan

--- 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


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Message: 321

Date: Mon, 25 Jun 2001 01:15:15

Subject: Re: Hypothesis revisited

From: Paul Erlich

--- 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.


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Message: 322

Date: Mon, 25 Jun 2001 02:43:09

Subject: Re: Hypothesis revisited

From: Dave Keenan

--- 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


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Message: 323

Date: Mon, 25 Jun 2001 03:15:37

Subject: Re: Hypothesis revisited

From: 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.


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Message: 324

Date: Mon, 25 Jun 2001 03:24:43

Subject: 41 "miracle" and 43 tone scales

From: jpehrson@r...

> 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


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