tctmo!

Here's the doc...

http://lumma.org/tuning/tctmo/

...Herman, you're welcome to link to the Mizarian Porcupine
Overture at http://lumma.org/tuning/tctmo/mizarian.mp3. Or
if you have a url for it elsewhere, I can use that instead.

Paul, Glassic is at http://lumma.org/tuning/tcmo/glassic.mp3.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Here's the doc...
>
> http://lumma.org/tuning/tctmo/

That should say some of the thinking, not all of it. Of course we
could get ambitious.

Outline

1-- Assume...

1.1-- Music involves repetition. Often, instead of exact
repetition, some things change while the rest stay the same.

1.1.1-- A theme played in a different mode keeps
generic
intervals (3rds, etc.) the same while pitches change.

1.1.1.1-- This is, in fact, only true for
Rothenberg-proper scales, such as the familiar
diatonic scale in 12-tone equal temperament.

1.1.2-- A theme played in a different key keeps
absolute
intervals the same while pitches change.

1.1.2.1-- As Rothenberg efficiency, the more
we
must rely on rules such as those traditionally
enforced in Western tonal music to make key
changes recognizable.

This is scale theory and should go under the heading of scales,
rather than as an introduction to everything.

1.2-- Possible intervals between notes are to be taken from
some
fixed set of just or near-just intervals, in order to best
exploit the signal-processing capabilities of the human
hearing
system to deliver information to the listener.

Is the argument about signal-processing capabilities basic to the
thinking on tuning-math? There are various reasons one could put
forward for limiting your intervals to a fixed set. Moreover, you
have not discussed octave equivalence and you need to at this point.

2-- So, to build a scale, we take our chosen just intervals and
*stack*
them. This generates a lattice.

This drags scales into an area where they are not required. Moreover,
you have not defined your terms. Do the chosen elements generate a
group, and is this "stacking"? Where and how does the
lattice turn up?

2.1-- In the 3-limit we get a chain. In the 5-limit, we get a
planar lattice. 7-limit, we can use the face-centered cubic
lattice. And so on.

You are assuming octave equivalence without having mentioned it. This
should be telling us that one generator gives us an infinite chain,
which can be embedded as the integers into the
real numbers, two elements give us a free group of rank two,
embeddable in R2 as a lattice in various ways, and so forth. You
haven't explained where the fcc 7-limit lattice arises from;
you need to put an inner product/quadratic form/metric/norm (however
you want to define it) onto your space to make the 5-limit lattice be
A2 and the 7-limit lattice be A3 = D3. The "and so on" gets you into
new problems. Is any of this really basic?

3-- Eventually, we will run into pitches that are very close to
pitcheswe already have. The small intervals between such pairs of
pitches are called commas.

This makes it sound as if the lattices needed to be introduced to do
this, which is not the case. Groups are more basic, why not start
there?

3.1-- We create a "pun" if we use the same name ("Ab") for
both
notes in such a pair.

3.2-- We create a "comma pump" by writing a chord progression
whose starting and ending "tonic" involve a "pun". Every time
the chord progression is repeated, our pitch standard moves by
the comma involved. Or...

We can create...

4-- We can temper the comma(s) out! Doing so collapses the lattice.
If
we temper enough commas out, we arrive at a finite "block" -- a
section
of the lattice delimited by the commas.

Once again, I suggest you start from groups, not lattices.

5-- It seems that most scales which have been popular around the world
and throughout history correspond fairly well to temperaments where
very simple commas have been tempered out.

5.1-- The complexity of a comma may be measured by the
distance
on the lattice between the pitches that generate it. Thus,
simpler commas tend to delimit smaller blocks.

This makes no sense to me.

5.1.1-- The size of the denominator of a comma
provides
a good "heuristic" for its complexity (Paul Erlich).

5.2-- The complexity of a temperament may be defined as
notes/intervals.

Again, this makes no sense to me.

5.2.1-- The complexity of a temperament is closely
related to the complexity of the commas which define
it.

Not true, I'm afraid. The commas can be arbitarily horrible so long
as there is more than one of them.

>This is scale theory and should go under the heading of scales,
>rather than as an introduction to everything.

They are merely examples of repetition where some things remain
fixed and others change. I need this to explain stacking.

>Is the argument about signal-processing capabilities basic to
>the thinking on tuning-math?

You can hardly get more basic. Why are we concerned with error?
Error from what? What separates what we do here from all the
other kooked-out music theory out there? Acoustics.

>There are various reasons one could put forward for limiting
>your intervals to a fixed set.

Such as? And not any fixed set, specifically just intonation.

>Moreover, you have not discussed octave equivalence and you need
>to at this point.

Octaves are just part of the fixed set, part of JI.

>2-- So, to build a scale, we take our chosen just intervals and
>*stack* them. This generates a lattice.
>
>This drags scales into an area where they are not required.

The original title of this document was "Scales and Temperaments".
Its original intent was to explain what I consider to be one of the
more important realizations of our work -- that the difference
between scales and temperaments is largely semantic.

>Moreover, you have not defined your terms. Do the chosen elements
>generate a group, and is this "stacking"? Where and how does the
>lattice turn up?

The intended audience doesn't know or care what a group is. The
lattice generating business could probably benefit from an example
and/or diagram.

> 2.1-- In the 3-limit we get a chain. In the 5-limit, we get a
> planar lattice. 7-limit, we can use the face-centered cubic
> lattice. And so on.
>
>You are assuming octave equivalence without having mentioned it.

You can show the octaves and just bump everything up. Maybe I
should say something about that.

>This should be telling us that one generator gives us an infinite
>chain, which can be embedded as the integers into the real numbers,
>two elements give us a free group of rank two, embeddable in R2 as
>a lattice in various ways, and so forth.

Gene, this is a 'for complete idiots' type document. I wonder how
many here even know what the heck this means.

>you need to put an inner product/quadratic form/metric/norm (however
>you want to define it) onto your space to make the 5-limit lattice be
>A2 and the 7-limit lattice be A3 = D3. The "and so on" gets you into
>new problems. Is any of this really basic?

Certainly not worded like that. If you can furnish a for-dummies
version, I'm happy to add it. Then again, I suggested you add it to
your own website weeks ago.

I think stacking is quite basic, and familiar to musicians and
tuning list readers.

>3-- Eventually, we will run into pitches that are very close to
>pitches we already have. The small intervals between such pairs of
>pitches are called commas.
>
>This makes it sound as if the lattices needed to be introduced to do
>this, which is not the case. Groups are more basic, why not start
>there?

Groups aren't more basic to tuning list readers. I'd love to start
with groups, if I knew what they had to do with music.

> 5.1-- The complexity of a comma may be measured by the
> distance on the lattice between the pitches that generate it.
> Thus, simpler commas tend to delimit smaller blocks.
>
>This makes no sense to me.

I'm not sure how I could possibly rephrase this.

> 5.1.1-- The size of the denominator of a comma
> provides a good "heuristic" for its complexity
>
> 5.2-- The complexity of a temperament may be defined as
> notes/intervals.
>
>Again, this makes no sense to me.

5.2 is a definition. Notes one has over consonant intervals formed
between them.

5.1.1 is Paul's complexity heuristic. The size of the denominator
of a pitch approximates the taxicab distance from the origin to it.

> 5.2.1-- The complexity of a temperament is closely
> related to the complexity of the commas which define
> it.
>
>Not true, I'm afraid. The commas can be arbitarily horrible so long
>as there is more than one of them.

Eh? Again, this is the heuristic. It only works for one comma.
But it should be applicable in the multiple-comma, with the addition
of a concept called "straightness".

-Carl

> 1.1-- Music involves repetition. Often, instead of exact
> repetition, some things change while the rest stay the same.

Schenker's motto, appearing on the title page of Der Freie Satz, was:

Semper idem sed non eodem modo.

"Always the same, but not in the same way."

I like your outline Carl: very "Tractatus Logico Philosophicus". Maybe the
last entry should be: "Those notes that cannot be played should be passed
over in silence". :-)

Dante

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Is the argument about signal-processing capabilities basic to
> >the thinking on tuning-math?
>
> You can hardly get more basic. Why are we concerned with error?

To keep things in tune.

> >Moreover, you have not discussed octave equivalence and you need
> >to at this point.
>
> Octaves are just part of the fixed set, part of JI.

Mostly, we've been looking at octave-reduced consonance sets.

> The original title of this document was "Scales and Temperaments".
> Its original intent was to explain what I consider to be one of the
> more important realizations of our work -- that the difference
> between scales and temperaments is largely semantic.

It sometimes is (ciculating temperaments) but not always (regular
temperaments.)

> >Moreover, you have not defined your terms. Do the chosen elements
> >generate a group, and is this "stacking"? Where and how does the
> >lattice turn up?
>
> The intended audience doesn't know or care what a group is. The
> lattice generating business could probably benefit from an example
> and/or diagram.

They'd better, since you need groups to define lattices. A lattice is
a free abelian group of finite rank with additional structure, the
additional structure being an embedding into a normed real vector
space. It is better not to add additional structure unless you have a
use for it; that merely confuses things.

> >This should be telling us that one generator gives us an infinite
> >chain, which can be embedded as the integers into the real numbers,
> >two elements give us a free group of rank two, embeddable in R2 as
> >a lattice in various ways, and so forth.
>
> Gene, this is a 'for complete idiots' type document. I wonder how
> many here even know what the heck this means.

I don't think it is a for complete idiots subject.

> >you need to put an inner product/quadratic form/metric/norm
(however
> >you want to define it) onto your space to make the 5-limit lattice
be
> >A2 and the 7-limit lattice be A3 = D3. The "and so on" gets you
into
> >new problems. Is any of this really basic?
>
> Certainly not worded like that. If you can furnish a for-dummies
> version, I'm happy to add it. Then again, I suggested you add it to
> your own website weeks ago.

I've been adding more to my website; you should check it out.
Lattices are a good idea for another addition, along with a host of
other things.

> I think stacking is quite basic, and familiar to musicians and
> tuning list readers.

I've never heard of it, so I can't see how it is integral to thinking
here.

> Groups aren't more basic to tuning list readers. I'd love to start
> with groups, if I knew what they had to do with music.

The p-limit intervals form a group. Equal temperaments are groups.
Linear temperaments are groups. Groups and group homomorphisms are
everywhere on this list. They are absolutely basic.

> > 5.1-- The complexity of a comma may be measured by the
> > distance on the lattice between the pitches that generate it.
> > Thus, simpler commas tend to delimit smaller blocks.
> >
> >This makes no sense to me.
>
> I'm not sure how I could possibly rephrase this.

What do you mean by "the pitches that generate it"?

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >>>Is the argument about signal-processing capabilities basic to
>>>the thinking on tuning-math?
>>
>>You can hardly get more basic. Why are we concerned with error?
> > To keep things in tune.

There's only a brief mention of signal processing. It's a definition of "in tune".

> Mostly, we've been looking at octave-reduced consonance sets.

Octave equivalence should certainly be mentioned in there.

> They'd better, since you need groups to define lattices. A lattice is > a free abelian group of finite rank with additional structure, the > additional structure being an embedding into a normed real vector > space. It is better not to add additional structure unless you have a > use for it; that merely confuses things. No. A lattice, in this context is a "network of points which show the simple translation vectors on which a structure is based." (H. M. Rosenberg, "The Solid State", third edition pp2-3)

> I don't think it is a for complete idiots subject.

Yes, we should at least target fairly intelligent idiots. But there's no reason to assume knowledge of group theory.

> The p-limit intervals form a group. Equal temperaments are groups. > Linear temperaments are groups. Groups and group homomorphisms are > everywhere on this list. They are absolutely basic.

"A scale is a group"?

>>> 5.1-- The complexity of a comma may be measured by the >>> distance on the lattice between the pitches that generate it.
>>> Thus, simpler commas tend to delimit smaller blocks.
>>>
>>>This makes no sense to me.
>>
>>I'm not sure how I could possibly rephrase this.
> > What do you mean by "the pitches that generate it"?

A comma is an interval between two pitches.

Graham

Carl Lumma wrote:

> You can hardly get more basic. Why are we concerned with error?
> Error from what? What separates what we do here from all the
> other kooked-out music theory out there? Acoustics.

I prefer to make my assumptions clear and leave acoustics out of it. It's then up to the reader whether or not they agree with it.

>>There are various reasons one could put forward for limiting
>>your intervals to a fixed set.
> > Such as? And not any fixed set, specifically just intonation.

No, the way I work it can be any fixed set of pitches. That's why I don't like these heuristics which tie it all to ratios for no good reason.

"Straightness" is another thing I never understood.

Graham

>Schenker's motto, appearing on the title page of Der Freie Satz, was:
>
>Semper idem sed non eodem modo.
>
>"Always the same, but not in the same way."

I just learned what Shenkerian analysis was back in January.
Totally cool.

>I like your outline Carl: very "Tractatus Logico Philosophicus".

Yeah, the numbering scheme. It was originally just bullets,
but I figured this would allow people to reference a section
without having to copy-and-paste it.

-Carl

>> You can hardly get more basic. Why are we concerned with error?
>> Error from what? What separates what we do here from all the
>> other kooked-out music theory out there? Acoustics.
>
>I prefer to make my assumptions clear and leave acoustics out of it.
>It's then up to the reader whether or not they agree with it.

Everything we do starts from the fundamental assumption of JI.
Or so it seems to me.

>>>There are various reasons one could put forward for limiting
>>>your intervals to a fixed set.
>>
>> Such as? And not any fixed set, specifically just intonation.
>
>No, the way I work it can be any fixed set of pitches. That's why I
>don't like these heuristics which tie it all to ratios for no good
>reason.

Za?

>"Straightness" is another thing I never understood.

It has to do with the angle between the commas. If A and B are
commas that vanish, and a and b are their lattice points, then the
interval C between a and b also vanishes. The thing is, A and B
could both be simple, but if the angle between them is wide, C could
still be complex. So you have to account for this in a heuristic.

-Carl

>> >Is the argument about signal-processing capabilities basic to
>> >the thinking on tuning-math?
>>
>> You can hardly get more basic. Why are we concerned with error?
>
>To keep things in tune.

And why do we care about that?

>> >Moreover, you have not discussed octave equivalence and you need
>> >to at this point.
>>
>> Octaves are just part of the fixed set, part of JI.
>
>Mostly, we've been looking at octave-reduced consonance sets.

Octaves are special, but there's no need to reflect this in a
theory of temperaments, other than weighting the error and/or
complexity functions. They're just like any other consonance
to be mapped.

>> The original title of this document was "Scales and Temperaments".
>> Its original intent was to explain what I consider to be one of the
>> more important realizations of our work -- that the difference
>> between scales and temperaments is largely semantic.
>
>It sometimes is (ciculating temperaments) but not always (regular
>temperaments.)

My plan is to paint circulating temperaments as small perturbations
of regular temperaments, not significant in the big-picture sense.

>> The intended audience doesn't know or care what a group is. The
>> lattice generating business could probably benefit from an example
>> and/or diagram.
>
>They'd better, since you need groups to define lattices. A lattice is
>a free abelian group of finite rank with additional structure, the
>additional structure being an embedding into a normed real vector
>space. It is better not to add additional structure unless you have a
>use for it; that merely confuses things.

It's jargon that's confusing.

If one can draw a lattice it should be sufficient to understand
this, without a rigorous definition.

>> >This should be telling us that one generator gives us an infinite
>> >chain, which can be embedded as the integers into the real numbers,
>> >two elements give us a free group of rank two, embeddable in R2 as
>> >a lattice in various ways, and so forth.
>>
>> Gene, this is a 'for complete idiots' type document. I wonder how
>> many here even know what the heck this means.
>
>I don't think it is a for complete idiots subject.

If you can't communicate with musicians and composers, what's the
point? You don't have to communicate everything to them. You can
work at another level. But at the end of the day, you should be
able to explain the big picture.

>> I think stacking is quite basic, and familiar to musicians and
>> tuning list readers.
>
>I've never heard of it, so I can't see how it is integral to
>thinking here.

Most musicians are familiar with structures like the circle of
fifths.

>> Groups aren't more basic to tuning list readers. I'd love to
>> start with groups, if I knew what they had to do with music.
>
>The p-limit intervals form a group. Equal temperaments are groups.
>Linear temperaments are groups. Groups and group homomorphisms are
>everywhere on this list. They are absolutely basic.

I hope you can show this on your web site.

I think I understand the mathworld definitions for group and
homomorphism, but I'm not sure of the alternatives. That is, basic
ratio arithmetic, which most folks on the tuning list already know,
seems to fail without a lot of this stuff. I'm not aware of all
the alternatives and their consequences, but many times when I do a
text substitution on the jargon in one of your math things it winds
up sounding totally obvious. Certainly, it doesn't end up sounding
like something that would lead to the results you get. Either I
misunderstand the definitions, you're leaving something out, or I
just don't see the deep consequences of the ensemble of definitions.

>> > 5.1-- The complexity of a comma may be measured by the
>> > distance on the lattice between the pitches that generate it.
>> > Thus, simpler commas tend to delimit smaller blocks.
>> >
>> >This makes no sense to me.
>>
>> I'm not sure how I could possibly rephrase this.
>
>What do you mean by "the pitches that generate it"?

You're right, this is bad. "Define" is maybe better than "generate".
Another way to say it might be "distance on the lattice between the
comma and the origin". Currently taking suggestions for even better
ways to say it. No jargon allowed.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:

> > They'd better, since you need groups to define lattices. A
lattice is
> > a free abelian group of finite rank with additional structure,
the
> > additional structure being an embedding into a normed real vector
> > space. It is better not to add additional structure unless you
have a
> > use for it; that merely confuses things.
>
> No. A lattice, in this context is a "network of points which show
the
> simple translation vectors on which a structure is based." (H. M.
> Rosenberg, "The Solid State", third edition pp2-3)

The lattices Carl were talking about were honest mathematical
lattices, and hence groups. Or so I think, anyway. Anyway, this is
the important case which gets the work done in most contexts, but the
work is simply abelian groups at work.

>
> > I don't think it is a for complete idiots subject.
>
> Yes, we should at least target fairly intelligent idiots. But
there's
> no reason to assume knowledge of group theory.

Why? Group theory is what works. We don't need anything complicated,
just some very basic stuff. Then, of course, I want to toss some
multilinear algebra into the works...

> > The p-limit intervals form a group. Equal temperaments are
groups.
> > Linear temperaments are groups. Groups and group homomorphisms
are
> > everywhere on this list. They are absolutely basic.
>
> "A scale is a group"?

Nope.

Carl Lumma wrote:

> Everything we do starts from the fundamental assumption of JI.
> Or so it seems to me.

The CGI won't do it yet, but my Python and OCaml modules are quite capable of working with an arbitrary set of prime intervals. The test script finds linear temperaments for Rayleigh's tubulong formula, as given in one of Brian McLaren's Xenharmonikon articles.

The "prime intervals" are simply a minimal set of intervals required to construct every interval in the scale. For prime-limit JI, these are the logarithms of the relevant prime numbers (modulo the log of 2 for octave equivalence). In terms of group theory, they're the generators of a finitely generated group and provide a homomorphism into the reals.

The only special treatement given to JI is that the prime intervals are supplied for you, there's a formula for expressing any ratio in terms of them and you can get a list of consonances that correspond to an odd limit.

> Za?

Is that supposed to mean something?

>>"Straightness" is another thing I never understood.
> > It has to do with the angle between the commas. If A and B are
> commas that vanish, and a and b are their lattice points, then the
> interval C between a and b also vanishes. The thing is, A and B
> could both be simple, but if the angle between them is wide, C could
> still be complex. So you have to account for this in a heuristic.

So it's an angle on the lattice? It's something that would be nice to have (and the heuristic certainly won't work without) but I've never been able to calculate it.

Graham

Gene Ward Smith wrote:

> The lattices Carl were talking about were honest mathematical > lattices, and hence groups. Or so I think, anyway. Anyway, this is > the important case which gets the work done in most contexts, but the > work is simply abelian groups at work.

Carl made two comments about your original statement

"It's jargon that's confusing."

Which may be correct. "Modern Computer Algebra" manages to define these kind of lattices without using group theory. I think they are still groups by that definition, but integers under addition are a group. You wouldn't require group theory for any paper that added numbers, would you? That would be confusing. As our lattices (by analogy to crystallography) can be explained without group theory it would only be confusing to require group theory.

"If one can draw a lattice it should be sufficient to understand
this, without a rigorous definition."

So he's talking about a lattice as something you draw. If distance on a lattice is supposed to be measured along the bases, I don't think they're algebraic lattices anyway, as that isn't a valid inner product.

Me:
>>Yes, we should at least target fairly intelligent idiots. But > there's >>no reason to assume knowledge of group theory.

Gene:
> Why? Group theory is what works. We don't need anything complicated, > just some very basic stuff. Then, of course, I want to toss some > multilinear algebra into the works...

We shouldn't assume knowledge because most people don't have it! Why do you have so much difficulty with that idea? It should be possible to understand our ideas with high school mathematics, and group theory is way beyond that.

If you want to note the connection with group theory as well, that's fine. But we shouldn't pollute our simple explanations with an infrastructure that most people don't know and don't want to know.

Graham

>The only special treatement given to JI is that the prime intervals
>are supplied for you, there's a formula for expressing any ratio in
>terms of them and you can get a list of consonances that correspond
>to an odd limit.

Ok, ok. But this is a caveat. We've got JI in our brains and one
can only go so far from it.

>> Za?
>
>Is that supposed to mean something?

It means "?".

>>>"Straightness" is another thing I never understood.
>>
>> It has to do with the angle between the commas. If A and B are
>> commas that vanish, and a and b are their lattice points, then the
>> interval C between a and b also vanishes. The thing is, A and B
>> could both be simple, but if the angle between them is wide, C could
>> still be complex. So you have to account for this in a heuristic.
>
>So it's an angle on the lattice? It's something that would be nice
>to have (and the heuristic certainly won't work without) but I've
>never been able to calculate it.

The heuristic works fine without it for linear temperaments. But
for more commas, one needs some way of taking into acount the
'difference vector(s)'. Straightness is Paul's idea, but I'm not
sure he ever suggested a way to calculate it. Maybe he would like
to say something here.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Here's the doc...
>
> http://lumma.org/tuning/tctmo/

rather than being "what's been going on on the tuning-math list",
it's really quite a small part, and virtually all of it dates back to
pre-tuning-math-list days.

already i see falsitudes:

> 1.1.1-- A theme played in a different mode keeps
>generic
> intervals (3rds, etc.) the same while pitches change.

> 1.1.1.1-- This is, in fact, only true for
> Rothenberg-proper scales, such as the familiar
> diatonic scale in 12-tone equal temperament.

the pythagorean diatonic is improper but would seem to have the
property you're trying to describe. so would blackjack . . .

> ...Herman, you're welcome to link to the Mizarian Porcupine
> Overture at http://lumma.org/tuning/tctmo/mizarian.mp3. Or
> if you have a url for it elsewhere, I can use that instead.
>
> Paul, Glassic is at http://lumma.org/tuning/tcmo/glassic.mp3.

thanks! the part that sounds vaguely like a repeating mixolydian I-
VII-VI-V progression is the part that uses the 7-tone porcupine scale.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Here's the doc...
>
> http://lumma.org/tuning/tctmo/

what happened to the corrections/reactions i already posted??

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> You can hardly get more basic. Why are we concerned with error?
> >> Error from what? What separates what we do here from all the
> >> other kooked-out music theory out there? Acoustics.
> >
> >I prefer to make my assumptions clear and leave acoustics out of
it.
> >It's then up to the reader whether or not they agree with it.
>
> Everything we do starts from the fundamental assumption of JI.
> Or so it seems to me.

is this the same "we" that didn't understand gene's initial keyboard
proposal? stop speaking for everyone.

>
> >"Straightness" is another thing I never understood.
>
> It has to do with the angle between the commas. If A and B are
> commas that vanish, and a and b are their lattice points, then the
> interval C between a and b also vanishes. The thing is, A and B
> could both be simple, but if the angle between them is wide, C could
> still be complex. So you have to account for this in a heuristic.
>
> -Carl

carl, this is far less important than the opposite case, where A and
B are very complex but C is very simple. C can't be any longer than
length (A) + length (B), but it can be arbitrarily shorter than
either.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> If one can draw a lattice it should be sufficient to understand
> this, without a rigorous definition.

graham's definition was far better than gene's, anyway.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The heuristic works fine without it for linear temperaments. But
> for more commas, one needs some way of taking into acount the
> 'difference vector(s)'. Straightness is Paul's idea, but I'm not
> sure he ever suggested a way to calculate it. Maybe he would like
> to say something here.
>
> -Carl

i tried my darndest here to get gene, kees, and others to help me
figure out which metric the heuristic implies, so that i could
generalize it to higher dimensions, but with no luck.

is the inner product defined over a triangular-taxicab metric?

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

> So it's an angle on the lattice? It's something that would be nice
to
> have (and the heuristic certainly won't work without) but I've
never
> been able to calculate it.

If we have a Euclidean lattice, we also have a corresponding
quadratic and bilinear form. The bilinear form gives us the dot
product, and hence angles. The question now becomes, what lattice are
we talking about, and is it Euclidean?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
>
> > So it's an angle on the lattice? It's something that would be
nice
> to
> > have (and the heuristic certainly won't work without) but I've
> never
> > been able to calculate it.
>
> If we have a Euclidean lattice, we also have a corresponding
> quadratic and bilinear form. The bilinear form gives us the dot
> product, and hence angles. The question now becomes, what lattice
are
> we talking about, and is it Euclidean?

we haven't yet been able to pin down the tempering rules on which the
error heuristic works, though it seems we must use the triangular-
taxicab metric on kees van prooijen's lattice to get the complexity
heuristic to work.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > If one can draw a lattice it should be sufficient to understand
> > this, without a rigorous definition.
>
> graham's definition was far better than gene's, anyway.

It depends on whether you are talking to a mathematician or not.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> is the inner product defined over a triangular-taxicab metric?

That's a normed vector space, but for a (positive definite) inner
product, what you get is a standard Euclidean vector space.

Paul Erlich wrote:

> the pythagorean diatonic is improper but would seem to have the > property you're trying to describe. so would blackjack . . .

I don't agree that the Pythagorean diatonic is improper. The two sizes of tritone are too close for most listeners to order them correctly when disjoint. And I don't agree that Blackjack has the property of diatonic modulation.

Now Carl, I don't think Part III of Rothenberg's series should be in the references. All you consider is propriety and efficiency.

The result of tempering out all commas can be thought of geometrically as collapsing the (hyper)plane into a (hyper)torus. Periodicity blocks are different because no commas are tempered out, so one block is different to another.

When you talk about lattice distance, you should say how the distance is measured.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > is the inner product defined over a triangular-taxicab metric?
>
> That's a normed vector space, but for a (positive definite) inner
> product, what you get is a standard Euclidean vector space.

i must have phrase my question incorrectly.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > the pythagorean diatonic is improper but would seem to have the
> > property you're trying to describe. so would blackjack . . .
>
> I don't agree that the Pythagorean diatonic is improper. The two
sizes
> of tritone are too close for most listeners to order them correctly
when
> disjoint.

how about the 17-equal diatonic?

> And I don't agree that Blackjack has the property of diatonic
> modulation.

where does it fail?

> The result of tempering out all commas can be thought of
geometrically
> as collapsing the (hyper)plane into a (hyper)torus.

that's what i told carl in my initial reply.

Paul Erlich wrote:

> how about the 17-equal diatonic?

I don't know. But Pythagorean's too marginal.

>>And I don't agree that Blackjack has the property of diatonic >>modulation.
> > where does it fail?

The large and small scale steps are so different that if the pattern changes a melody is qualitatively different.

> that's what i told carl in my initial reply.

If that was to "tcmo!" I don't have it.

Graham

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > is the inner product defined over a triangular-taxicab metric?
> >
> > That's a normed vector space, but for a (positive definite) inner
> > product, what you get is a standard Euclidean vector space.
>
> i must have phrase my question incorrectly.

The short answer would be "no".

> If that was to "tcmo!" I don't have it.

Or even "tctmo!"

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > how about the 17-equal diatonic?
>
> I don't know. But Pythagorean's too marginal.
>
> >>And I don't agree that Blackjack has the property of diatonic
> >>modulation.
> >
> > where does it fail?
>
> The large and small scale steps are so different that if the
pattern
> changes a melody is qualitatively different.

that seems subjective. how about the 17-equal diatonic case? you
don't know?

>
> > that's what i told carl in my initial reply.
>
> If that was to "tcmo!" I don't have it.

no, it was "review . . ."

Carl Lumma wrote:

> Ok, ok. But this is a caveat. We've got JI in our brains and one
> can only go so far from it.

The theory shouldn't have to rely on empirical details about the human brain.

> It means "?".

The heuristics involves numerators and denominators, and interval vectors in general don't have numerators and denominators. But, as I saw one of them derived, it's only an approximation to something that you can calculate directly for all interval vectors as I use them. So there's an avoidable loss of generality.

> The heuristic works fine without it for linear temperaments. But
> for more commas, one needs some way of taking into acount the
> 'difference vector(s)'. Straightness is Paul's idea, but I'm not
> sure he ever suggested a way to calculate it. Maybe he would like
> to say something here.

For 5-limit linear temperaments.

Graham

Paul Erlich wrote:

> we haven't yet been able to pin down the tempering rules on which the > error heuristic works, though it seems we must use the triangular-
> taxicab metric on kees van prooijen's lattice to get the complexity > heuristic to work.

The error heuristic I use is size/complexity where size is the magnitude of the pitch difference and complexity is the smallest number of intervals in the relevant odd limit that make up the comma. I think both are valid metrics, but complexity isn't an inner product. You could approximate it by straight line distance on a triangular lattice or rectangular octave-specific lattice.

The error measured is exactly the worst case of the worst tuning of any interval in the relevant limit for an optimized temperament with the given comma.

What's special about kees's lattice? Triangular-taxicab distances on an fcc lattice won't give the right results for 9-limit intervals. Weighting the 3 direction as half the size of the others will approximate them (probably as well as your heuristic). Straight line distances with scaled axes won't be much different, and are probably the best bet as they are inner products, so we can get angles from them.

Graham

Paul Erlich wrote:

> that seems subjective. how about the 17-equal diatonic case? you > don't know?

Yes, it's subjective.

> no, it was "review . . ."

Right!

Graham

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

> I think
> both are valid metrics, but complexity isn't an inner product.

who said it was?

> What's special about kees's lattice? Triangular-taxicab distances
on an
> fcc lattice won't give the right results for 9-limit intervals.

that's where "wormholes" come in.

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

> The error heuristic I use is size/complexity where size is the
magnitude
> of the pitch difference

What pitch difference? A comma has a size, namely itself; we can take
the log of that also, of course.

and complexity is the smallest number of
> intervals in the relevant odd limit that make up the comma.

A comma is made up of intervals? I thought it *was* an interval.

I think
> both are valid metrics, but complexity isn't an inner product.

If you want an inner product, what about the one I use to define
geometric complexity?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> If you want an inner product, what about the one I use to define
> geometric complexity?

probably not a bad choice, at least for this community at this point
in time.

i'm amazed at your inability to understand graham, though!

Gene Ward Smith wrote:

> What pitch difference? A comma has a size, namely itself; we can take > the log of that also, of course.

The size of a comma depends on the metric you apply. An interval is the ratio of two frequencies, or the difference between two pitches, where pitch is the logarithm of frequency. To get the right error heuristic, the error has to be as a pitch difference.

> A comma is made up of intervals? I thought it *was* an interval.

Yes, an interval is an element of an abelian group, remember? So intervals can be produced from other intervals. The interval 81:80, for example, can be made up of 3:2 * 3:2 * 3:2 * 3:2 * 1:5. But it's simpler to break it down into 3:5 * 3:4 * 3:2 * 3:2. That involves 4 intervals in the 5-limit, so 81:80 has a 5-limit complexity of 4.

> If you want an inner product, what about the one I use to define > geometric complexity?

Yes, that'll do. But so would a standard dot product of octave-specific vectors, scaled by the size of the prime intervals. That would be simpler, and give much the same result for sufficiently small (as a pitch difference) commas.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > What pitch difference? A comma has a size, namely itself; we can
take
> > the log of that also, of course.
>
> The size of a comma depends on the metric you apply.

Ah. We're talking norms, then.

>> Here's the doc...
>>
>> http://lumma.org/tuning/tctmo/
>
>what happened to the corrections/reactions i already posted??

They were added.

-C.

>> Everything we do starts from the fundamental assumption of JI.
>> Or so it seems to me.
>
>is this the same "we" that didn't understand gene's initial keyboard
>proposal?

I dunno. It didn't seem like Graham understood it, until I created
a diagram of it. I understood it at first, but then changed my
mind after Gene pointed us to a diagram that bore absolutely no
resemblance to what he described.

>stop speaking for everyone.

>> Or so it seems to me.

>> It has to do with the angle between the commas. If A and B are
>> commas that vanish, and a and b are their lattice points, then the
>> interval C between a and b also vanishes. The thing is, A and B
>> could both be simple, but if the angle between them is wide, C could
>> still be complex. So you have to account for this in a heuristic.
>>
>> -Carl
>
>carl, this is far less important than the opposite case, where A and
>B are very complex but C is very simple. C can't be any longer than
>length (A) + length (B), but it can be arbitrarily shorter than
>either.

Ah, ok.

-Carl

>It depends on whether you are talking to a mathematician or not.

Gene,

It strikes me as quite possible that group theory is a better
basis from which to explain this stuff, but can a group theory
crash course be fit into a short document?

-Carl

>Now Carl, I don't think Part III of Rothenberg's series should
>be in the references. All you consider is propriety and efficiency.

Yeah, I was going to take it out.

>The result of tempering out all commas can be thought of geometrically
>as collapsing the (hyper)plane into a (hyper)torus. Periodicity blocks
>are different because no commas are tempered out, so one block is
>different to another.

The way the document is currently set up, tempered things can still
be called blocks. Is that unacceptable to anyone?

>When you talk about lattice distance, you should say how the distance
>is measured.

An earlier version mentioned "taxicab", but I pulled it, because
Gene's metric isn't taxicab.

Paul, I just noticed you said the complexity heuristic doesn't work
but with taxicab on a kees lattice. What are the failing cases?

-Carl

>> Ok, ok. But this is a caveat. We've got JI in our brains and one
>> can only go so far from it.
>
>The theory shouldn't have to rely on empirical details about the
>human brain.

This is about music, right?

>> It means "?".
>
>The heuristics involves numerators and denominators, and interval
>vectors in general don't have numerators and denominators. But, as I
>saw one of them derived, it's only an approximation to something that
>you can calculate directly for all interval vectors as I use them. So
>there's an avoidable loss of generality.

What's an interval vector?

>> The heuristic works fine without it for linear temperaments. But
>> for more commas, one needs some way of taking into acount the
>> 'difference vector(s)'. Straightness is Paul's idea, but I'm not
>> sure he ever suggested a way to calculate it. Maybe he would like
>> to say something here.
>
>For 5-limit linear temperaments.

Right; sorry.

-Carl

>The error heuristic I use is size/complexity where size is the
>magnitude of the pitch difference

That sounds the same as Paul's heuristic.

>and complexity is the smallest
>number of intervals in the relevant odd limit that make up the
>comma.

This sounds like taxicab distance. Paul uses d, but d is
supposed to be like taxicab distance (at least, on the right
lattice)...

>The error measured is exactly the worst case of the worst tuning
>of any interval in the relevant limit for an optimized temperament
>with the given comma.

You seem to like worst error more than average error. Any reason?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >It depends on whether you are talking to a mathematician or not.
>
> Gene,
>
> It strikes me as quite possible that group theory is a better
> basis from which to explain this stuff, but can a group theory
> crash course be fit into a short document?

The kind of groups we are most interested in are free abelian groups
of finite rank, and they can be equated with row vectors of integers.

>> It strikes me as quite possible that group theory is a better
>> basis from which to explain this stuff, but can a group theory
>> crash course be fit into a short document?
>
>The kind of groups we are most interested in are free abelian groups
>of finite rank, and they can be equated with row vectors of integers.

I don't see how this answers my question, but I understand it, at
least on some level. But, once again, isn't this stuff assumed
by the basic ratio math everybody already uses? Wouldn't anything
but an abelian group be catastrophic? If so, I don't see why it's
so important.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> It strikes me as quite possible that group theory is a better
> >> basis from which to explain this stuff, but can a group theory
> >> crash course be fit into a short document?
> >
> >The kind of groups we are most interested in are free abelian
groups
> >of finite rank, and they can be equated with row vectors of
integers.
>
> I don't see how this answers my question, but I understand it, at
> least on some level. But, once again, isn't this stuff assumed
> by the basic ratio math everybody already uses? Wouldn't anything
> but an abelian group be catastrophic? If so, I don't see why it's
> so important.

One important reason is that you want the mechanics of mappings
(homomorphisms) to be available. There's an adage in mathematics that
the morphisms are more important that the objects, and that holds
here. It's not very exciting to know that an equal temperament
considered as a group is isomorphic to the integers; it really
requires a mapping (telling us, at the very least, where the octave
is) to make any sense. The mappings also have kernels, which leads us
to commas.

Carl Lumma wrote:

> I don't see how this answers my question, but I understand it, at
> least on some level. But, once again, isn't this stuff assumed
> by the basic ratio math everybody already uses? Wouldn't anything
> but an abelian group be catastrophic? If so, I don't see why it's
> so important.

But that's why it's important! Groups are intended to generalize arithmetic operations. The positive rationals under multiplication do constitute a group. Unless you stay with JI of harmonic timbres, you need to consider a more general case. Tempered intervals can be added as well, but they don't work with ratio math. Abelian groups constitute a good level of abstraction that wasn't invented for us.

There are some other properties of rational numbers we still need. The number itself represents the frequency ratio between two notes, so we need a mapping from notes to real numbers representing frequency or pitch. For JI and regular temperaments, this mapping is a homomorphism of intervals. The complexity of a ratio gives some idea of dissonance, but this doesn't really generalize outside of JI, and is one of the things we're currently discussing. The difference between rationals gives the beat ratio. In the general case, that's a function of frequencies.

The thing is, although group theory is applicable, I don't know if it's useful enough in this context for people who don't already know it. The same things can be expressed using matrix algebra, which also allows us to solve a set of commas to get a temperament. You have to know about either matrix adjoints or wedge products to do this, but there's nothing so far that requires group theory. It only provides a language for expressing certain generalizations.

Graham

Me:
>>The error heuristic I use is size/complexity where size is the
>>magnitude of the pitch difference

Carl:
> That sounds the same as Paul's heuristic.

I think Dave used it first. Paul's heuristic is an approximation assuming the comma to be a ratio.

>>and complexity is the smallest
>>number of intervals in the relevant odd limit that make up the
>>comma.
> > This sounds like taxicab distance. Paul uses d, but d is
> supposed to be like taxicab distance (at least, on the right
> lattice)...

It'd be a taxicab distance on the right lattice, assuming the roads form an n-dimensional lattice triangular in cross section with wormholes. What's d?

> You seem to like worst error more than average error. Any reason?

In this case, worst error is what the heuristic works for. I'm using RMS error in software because it's easier to optimize for.

Graham

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

> The thing is, although group theory is applicable, I don't know if
it's
> useful enough in this context for people who don't already know it.

Academic theorists have taken up groups, so it will not be utterly
unfamiliar to people coming from that quarter.

Carl Lumma wrote:

> This is about music, right?

It's about tuning systems, which may be used to make music. There are already tuning systems derived from inharmonic timbres. A general method of finding temperaments will be most useful when considering timbres that haven't been considered before.

> What's an interval vector?

A pitch difference expressed as a list of integers, which is much the same as a free albelian group.

Graham

Gene Ward Smith wrote:

> Academic theorists have taken up groups, so it will not be utterly > unfamiliar to people coming from that quarter.

So we are intending this "for dummies"? ;-)

It wouldn't do any harm to point out that directed intervals (remember the "directed"; I usually don't, and it is important) form an abelian group and the specific representation we use is of a free abelian group. If people don't understand that, they can ignore it -- possibly it can be in a footnote.

Beyond that, we're already using the term "generator". Homomorphisms are important, and we may as well call them such. But we'll need to explain them in musical terms. Something like "each just interval has a counterpart in the temperament, and adding intervals should give the same result whether the mapping occurs before or after the addition". The link with unison vectors and the kernel of a homomorphism can be in a footnote.

Equal temperaments as cyclic groups is in Balzano (1980) so we'll have to reference it!

Otherwise, what group theory do you want?

Graham

Gene Ward Smith wrote:

> Ah. We're talking norms, then.

The size of an interval as a pitch difference is a norm but not a p-norm. In general terms, we're working with directed intervals so we need the sign as well. That isn't a norm. It is really the physics definition of a vector -- a magnitude and a direction. The norm gives us the magnitude. If the direction is the one-dimensional equivalent of an angle, what we need is a metric, the way I was taught:

"... the notions of /length/ and /angle/ are represented by a /metric/." (Ian D. Lawrie, "A Unified Grand Tour of Theoretical Physics", pp. 29-30)

For a regular tuning, we need a homomorphism into R, which can be interpreted as a pitch difference.

For a well temperament, the mapping has to be from notes to pitches. The size of an interval depends on what note it starts on, so it isn't a homomorphism.

Graham

>>>and complexity is the smallest
>>>number of intervals in the relevant odd limit that make up the
>>>comma.
>>
>> This sounds like taxicab distance. Paul uses d, but d is
>> supposed to be like taxicab distance (at least, on the right
>> lattice)...
>
>It'd be a taxicab distance on the right lattice, assuming the roads form
>an n-dimensional lattice triangular in cross section with wormholes.
>What's d?

Denominator.

-Carl

>> What's an interval vector?
>
>A pitch difference expressed as a list of integers, which is much the
>same as a free albelian group.

So you need some kind of space. How do you 'factor' irrational
intervals to get this space?

-Carl

>Homomorphisms are important, and we may as well call them such. But
>we'll need to explain them in musical terms. Something like "each
>just interval has a counterpart in the temperament, and adding
>intervals should give the same result whether the mapping occurs
>before or after the addition".

Hey, that's good. Mind if I use it?

-Carl

> Hey, that's good. Mind if I use it?

Yes, of course.

Carl Lumma wrote:

> So you need some kind of space. How do you 'factor' irrational
> intervals to get this space?

You can take the most prominent partials, relative to the fundamental, as a set of prime intervals, to take the place of prime numbers for JI of harmonic timbres. Sharp minima in the dissonance curve will usually be simple combinations of these intervals. If you don't have dissonance curve, taking a cross set will do.

For example, on p.170 of Sethares' magnum opus, a typical spectrum for Swstigitha sarons is given as f, 2.76f, 4.72f and 5.92f. I'll take 2.76 as the equivalence interval and use a standard two-dimensional cross set (the equivalent of the 5-limit)

>>> import temper, math
>>> swastigitha = temper.PrimeDiamond(2)
>>> swastigitha.primes = [math.log(x)/math.log(2.76) for x in
(4.72, 5.92)]

So that's it, a set of consonances is defined according to the empirical spectrum. Temperaments can be generated from it like any other set of consonances

>>> [et.basis[0] for et in temper.getLimitedETs(swastigitha)]
[1, 4, 5, 9, 11, 13, 15, 17, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32]
>>> temper.getLinearTemperaments(temper.getLimitedETs(swastigitha))[0]

11/36, 305.7 milli-equivalence generator

basis:
(1.0, 0.30573972381117814)

mapping by period and generator:
[(1, 0), (0, 5), (-1, 9)]

mapping by steps:
[(23, 13), (35, 20), (40, 23)]

highest interval width: 9
complexity measure: 9 (10 for smallest MOS)
highest error: 0.000170 (0.170 milli-equivalences)
unique

The equivalence interval here is 1757.6 cents, so the generator of that linear temperament is 537.4 cents, and the worst error 0.3 cents.

Alternatively, using a regular octave as the equivalence interval, to fit better with other instruments

>>> swas_oct = temper.PrimeDiamond(3)
>>> swas_oct.primes = [temper.log2(x) for x in (2.76, 4.72, 5.92)]
>>> [et.basis[0] for et in temper.getLimitedETs(swas_oct)]
[4, 5, 8, 9, 12, 13, 17, 19, 21, 22, 28, 30, 32, 34, 37, 39, 40, 41, 43, 46]
>>> temper.getLinearTemperaments(temper.getLimitedETs(swas_oct))[0]

2/7, 101.0 milli-equivalence generator

basis:
(0.33333333333333331, 0.10099919983161132)

mapping by period and generator:
[(3, 0), (5, -2), (7, -1), (8, -1)]

mapping by steps:
[(12, 9), (18, 13), (27, 20), (31, 23)]

highest interval width: 2
complexity measure: 6 (9 for smallest MOS)
highest error: 0.006523 (6.523 milli-equivalences)

101 millioctaves is 121.2 cents.

Graham

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Here's the doc...
> >>
> >> http://lumma.org/tuning/tctmo/
> >
> >what happened to the corrections/reactions i already posted??
>
> They were added.
>
> -C.

not very well.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >When you talk about lattice distance, you should say how the
distance
> >is measured.
>
> An earlier version mentioned "taxicab", but I pulled it, because
> Gene's metric isn't taxicab.
>
> Paul, I just noticed you said the complexity heuristic doesn't work
> but with taxicab on a kees lattice. What are the failing cases?

all the others! i mean, it's always a good heuristic, but it's exact
for kees' lattice . . .

I wrote:

> Alternatively, using a regular octave as the equivalence interval, to > fit better with other instruments
> > >>> swas_oct = temper.PrimeDiamond(3)
> >>> swas_oct.primes = [temper.log2(x) for x in (2.76, 4.72, 5.92)]
> >>> [et.basis[0] for et in temper.getLimitedETs(swas_oct)]
> [4, 5, 8, 9, 12, 13, 17, 19, 21, 22, 28, 30, 32, 34, 37, 39, 40, 41, 43, 46]
> >>> temper.getLinearTemperaments(temper.getLimitedETs(swas_oct))[0]
> > 2/7, 101.0 milli-equivalence generator

...

These timbres are used with a pelog tuning. The second best linear temperament does look a bit like pelog:

>>> temper.getLinearTemperaments(temper.getLimitedETs(swas_oct))[1]

2/9, 224.8 milli-equivalence generator

basis:
(1.0, 0.2247677727873047)

mapping by period and generator:
[(1, 0), (1, 2), (2, 1), (3, -2)]

mapping by steps:
[(5, 4), (7, 6), (11, 9), (13, 10)]

highest interval width: 4
complexity measure: 4 (5 for smallest MOS)
highest error: 0.015133 (15.133 milli-equivalences)

The generator is half of pelog's. To get pelog itself, the cutoff for equal temperments has to be raised to allow 7-equal in. Then, pelog narrowly makes the top 10:

>>> temper.getLinearTemperaments(temper.getLimitedETs(swas_oct, cutoff=0.6))[9]

4/9, 445.2 milli-equivalence generator

basis:
(1.0, 0.44517628148326549)

mapping by period and generator:
[(1, 0), (1, 1), (4, -4), (3, -1)]

mapping by steps:
[(7, 2), (10, 3), (16, 4), (18, 5)]

highest interval width: 5
complexity measure: 5 (7 for smallest MOS)
highest error: 0.019492 (19.492 milli-equivalences)
unique

The situation improves slightly when you add in some harmonic partials. With 3:1, and allowing for a bit of inconsistency, pelog is the fourth best.

>>> swas2 = temper.PrimeDiamond(4)
>>> swas2.primes = [temper.log2(x) for x in (2.76, 3, 4.72, 5.92)]
>>> temper.getLinearTemperaments(temper.getLimitedETs(swas2, cutoff=0.6), worstError=0.1)[3]

4/9, 440.3 milli-equivalence generator

basis:
(1.0, 0.44030328510322092)

mapping by period and generator:
[(1, 0), (1, 1), (2, -1), (4, -4), (3, -1)]

mapping by steps:
[(7, 2), (10, 3), (11, 3), (16, 4), (18, 5)]

highest interval width: 5
complexity measure: 5 (7 for smallest MOS)
highest error: 0.025266 (25.266 milli-equivalences)

And with 3:1 and 5:1, it moves down to 7

>>> swas3 = temper.PrimeDiamond(5)
>>> swas3.primes = [temper.log2(x) for x in (2.76, 3.0, 4.72, 5.0, 5.92)]

>>> temper.getLinearTemperaments(temper.getLimitedETs(swas3, cutoff=0.6))[6]

7/16, 440.3 milli-equivalence generator

basis:
(1.0, 0.44030328510322092)

mapping by period and generator:
[(1, 0), (1, 1), (2, -1), (4, -4), (1, 3), (3, -1)]

mapping by steps:
[(9, 7), (13, 10), (14, 11), (20, 16), (21, 16), (23, 18)]

highest interval width: 7
complexity measure: 7 (9 for smallest MOS)
highest error: 0.025266 (25.266 milli-equivalences)

The scale recorded for this gamelan doesn't fit the theoretical large-small pattern for pelog very well either.

Graham

>> So you need some kind of space. How do you 'factor' irrational
>> intervals to get this space?
>
>You can take the most prominent partials, relative to the fundamental,
>as a set of prime intervals, to take the place of prime numbers for JI
>of harmonic timbres.

Ok, sure. I'll take out "just intonation" and replace it with "consonant
intervals".

-Carl

>>Schenker's motto, appearing on the title page of Der Freie Satz,
>>was: Semper idem sed non eodem modo.
>>
>>"Always the same, but not in the same way."
>
>I just learned what Shenkerian analysis was back in January.
>Totally cool.

On these lists I've often hinted at something like "parathesis
checking" (an algorithm that checks if there is a close-bracket
for every open bracket) for studying symmetrical melodies as
found in Mozart, Sousa, Joplin, etc. Kinda reminiscent of
Shenkerian analysis. Also, Boomsliter & Creel's "extended
reference".

The way I was shown, two different people might produce two
very different Shenkerian breakdowns of the same piece. I
wonder if anyone's tried to automate it in a deterministic
fashion?

-Carl

>rather than being "what's been going on on the tuning-math list",

What rather than "what's been going on on the tuning-math list"?

>> 1.1.1-- A theme played in a different mode keeps
>> generic intervals (3rds, etc.) the same while pitches
>> change.
>
>> 1.1.1.1-- This is, in fact, only true for
>> Rothenberg-proper scales, such as the familiar
>> diatonic scale in 12-tone equal temperament.
>
>the pythagorean diatonic is improper but would seem to have the
>property you're trying to describe. so would blackjack . . .

I suppose any scale would, if we define generic intervals as simply
being the intervals between consecutive scale degrees. We need
the assumption that listeners keep track of the relative sizes of
intervals to build a map of generic intervals. This is getting a
bit complicated, so I've axed the section.

-Carl

S. analysis is much more an art than a science. Large-scale features should
show up in anyone's graph but details will vary. Of course, two people could
write two different programs to automate it and produce different but
perhaps equally valid results. Even simple harmonic analysis of say, Brahms,
will vary from person to person due to strange spellings and ambiguity. Even
moreso with Wagner or Strauss. S. was also limited by his "German Tonal Art"
blinders (although Chopin was an "honorary German").

Dante

> -----Original Message-----
> From: Carl Lumma [mailto:ekin@lumma.org]
> Sent: Thursday, August 14, 2003 2:25 AM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Shenkerian analysis
>
>
> >>Schenker's motto, appearing on the title page of Der Freie Satz,
> >>was: Semper idem sed non eodem modo.
> >>
> >>"Always the same, but not in the same way."
> >
> >I just learned what Shenkerian analysis was back in January.
> >Totally cool.
>
> On these lists I've often hinted at something like "parathesis
> checking" (an algorithm that checks if there is a close-bracket
> for every open bracket) for studying symmetrical melodies as
> found in Mozart, Sousa, Joplin, etc. Kinda reminiscent of
> Shenkerian analysis. Also, Boomsliter & Creel's "extended
> reference".
>
> The way I was shown, two different people might produce two
> very different Shenkerian breakdowns of the same piece. I
> wonder if anyone's tried to automate it in a deterministic
> fashion?
>
> -Carl
>
>
>
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