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Message: 7175 Date: Fri, 01 Aug 2003 00:23:07 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Yes, I chased this down, and found a definition here: > > Yahoo groups: /tuning-math/message/5546 * [with cont.] > > Given that it is so straightforward, why do you exclude it from

your

> other listings?

It follows immediately from the definition.

> So, I've got as far as converting that into Python: > > import math, temper > > def complexity5(u, primes=None): > primes = primes or temper.primes[:u.maxBasis()] > return math.log(2) * math.sqrt( > primes[0]**2 * u[1,]**2 + > primes[0]**2 * u[1,] * u[2,] + > primes[1]**2 * u[2,]**2) > >

> >>If I had worked out your numbering rule, I've forgotten it now. > >> > >>Every time you try to explain something, you bring in more jargon

> > > > terms > >

> >>that I don't understand (I can't speak for anybody else). > >> > >>The word "metric" in particular is something that's important but

> > > > you > >

> >>haven't defined.

> > > > > > I've posted this before; it's standard math: > > > > Metric -- from MathWorld * [with cont.]

> > *That's* standard math, yes, but it doesn't say anything about

applying

> a metric to an exterior algebra. And in the geometric complexity

definiton:

> > Yahoo groups: /tuning-math/message/4533 * [with cont.] > > you give what is, as far as I can work out, a function of a single > rational number as the "metric".

A single rational number, in terms of 5-limit note-classes, is a two- dimensional vector. Does that explain it for you?

Message: 7176 Date: Fri, 01 Aug 2003 12:02:23 Subject: Re: Creating a Temperment /Comma From: Graham Breed Me:

>>Given that it is so straightforward, why do you exclude it from

> your

>>other listings?

Gene:

> It follows immediately from the definition.

So it follows from a definition that nobody understands! (Well does anybody understand it? Speak up!) That isn't any use at all. Me:

>>Yahoo groups: /tuning-math/message/4533 * [with cont.] >> >>you give what is, as far as I can work out, a function of a single >>rational number as the "metric".

Gene:

> A single rational number, in terms of 5-limit note-classes, is a two- > dimensional vector. Does that explain it for you?

No, that makes even less sense. You seem to be suggesting that the two coefficients of an octave-equivalent 5-limit vector constitute the parameters for a metric. From where I'm sitting, that's gibberish. It means those two coefficients are supposed to be points in some abstract space. And to be symmetric, the metric would have to treat the two coefficients as interchangeable, so you certainly can't multiply one by log(3) and the other by log(5) like you do in geometric complexity. As for the g(x,x)=0 rule, that means the "distance" of 15:8 is zero, the same as a unison! And even if you could somehow (using a process you haven't explained) get geometric complexity from such a measure, it would only be defined in the 5-limit. That's a pervers way of getting a very simple result. I'm guessing that "note-classes" are something to do with octave equivalence -- there's another piece of jargon you haven't defined. Graham

Message: 7178 Date: Fri, 01 Aug 2003 15:35:02 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> Me:

> >>Given that it is so straightforward, why do you exclude it from

> > your

> >>other listings?

> > Gene:

> > It follows immediately from the definition.

> > So it follows from a definition that nobody understands! (Well

does

> anybody understand it? Speak up!) That isn't any use at all.

Up until now I had thought you understood geometric complexity. You remarked once that it might be better to use logs base two for the definition, which is true and insightful.

> Me:

> >>Yahoo groups: /tuning-math/message/4533 * [with cont.] > >> > >>you give what is, as far as I can work out, a function of a

single

> >>rational number as the "metric".

> > Gene:

> > A single rational number, in terms of 5-limit note-classes, is a

two-

> > dimensional vector. Does that explain it for you?

> > No, that makes even less sense. You seem to be suggesting that the

two

> coefficients of an octave-equivalent 5-limit vector constitute the > parameters for a metric. From where I'm sitting, that's

gibberish. It

> means those two coefficients are supposed to be points in some

abstract

> space.

Correct. And to be symmetric, the metric would have to treat the two

> coefficients as interchangeable, so you certainly can't multiply

one by

> log(3) and the other by log(5) like you do in geometric complexity.

Not correct. All that is required is that the matrix B(i,j) that you are summing over (as sum_{i,j} B(i,j) x_i x_j) be a symmetric matrix with real and positive eigenvalues. Here's the usual grab bag from Weisstein: Symmetric Bilinear Form -- from MathWorld * [with cont.] http://mathworld.wolfram.com/QuadraticForm.html * [with cont.] http://mathworld.wolfram.com/PositiveDefiniteQ... * [with cont.] Hermitian Inner Product -- from MathWorld * [with cont.] As

> for the g(x,x)=0 rule, that means the "distance" of 15:8 is zero,

the

> same as a unison!

Eh? We are looking at g([0,0], [1, 1]) to find the distance of 15/8 to 1. If we find the distance of 15/8 to itself, we get zero, which is correct. And even if you could somehow (using a process you

> haven't explained) get geometric complexity from such a measure, it > would only be defined in the 5-limit. That's a pervers way of

getting a

> very simple result.

No, because in other cases we use the metric induced on wedge products by the metric we started out from.

> I'm guessing that "note-classes" are something to do with octave > equivalence -- there's another piece of jargon you haven't defined.

I thought "note-classes" was standard music theory jargon for the equivalence classes defined by octave equivalence.

Message: 7179 Date: Sat, 02 Aug 2003 11:04:52 Subject: Re: Graham definitions for geometric complexity and badness From: Carl Lumma

>> No, no, Graham is a cool first name, and first names >> should be used when possible. :)

> >What's wrong with just calling it geometric complexity? You are going >to get this mixed up with Graham's complexity for linear temperaments.

It was you who called it that! But you're right. -Carl

Message: 7180 Date: Sat, 2 Aug 2003 11:14:27 Subject: Re: Graham definitions for geometric complexity and badnesss From: monz@xxxxxxxxx.xxx hi Gene,

> From: Carl Lumma [mailto:ekin@xxxxx.xxxx > Sent: Saturday, August 02, 2003 11:05 AM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: Re: [tuning-math] Re: Graham definitions for geometric > complexity and badness > >

> >> No, no, Graham is a cool first name, and first names > >> should be used when possible. :)

> > > > What's wrong with just calling it geometric complexity? > > You are going to get this mixed up with Graham's complexity > > for linear temperaments.

> > It was you who called it that! But you're right.

OK, i'll make a Dictionary entry for "geometric complexity". if you give me an opening paragraph describing what it is, i can just use the rest of your previous post for the definition. -monz

Message: 7181 Date: Sat, 02 Aug 2003 19:28:02 Subject: Re: Creating a Temperment /Comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >>>consistency is only defined for ets.

> >> > >>What Gene said about regular temperaments sounded > >>equivalent to me.

> > > >a definition of consistency for regular temperaments? > >doesn't seem possible, as they're infinite, so you can > >always find better and better approximations to anything.

> > Yahoo groups: /tuning-math/message/3330 * [with cont.]

umm . . . so?

Message: 7182 Date: Sat, 02 Aug 2003 22:52:34 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>> >>>consistency is only defined for ets.

>> >> >> >>What Gene said about regular temperaments sounded >> >>equivalent to me.

>> > >> >a definition of consistency for regular temperaments? >> >doesn't seem possible, as they're infinite, so you can >> >always find better and better approximations to anything.

>> >> Yahoo groups: /tuning-math/message/3330 * [with cont.]

> >umm . . . so?

So, doesn't sound like you can break consistency in regular temperaments! -Carl

Message: 7183 Date: Sat, 02 Aug 2003 00:45:00 Subject: Re: Creating a Temperment /Comma From: Graham Breed Gene Ward Smith wrote:

> Up until now I had thought you understood geometric complexity. You > remarked once that it might be better to use logs base two for the > definition, which is true and insightful.

No, I said I didn't understand it. I could tell from the examples that it involves logs of primes, and that any logarithms should be to base 2 in this context. In much the same way, I can see that the units depend on the codimension? of the wedgie. So an interval has complexity measured in octaves, but a linear temperament (or pair of commas) is in octaves^2. Would it make sense to take root so that everything's in octaves?

> Not correct. All that is required is that the matrix B(i,j) that you > are summing over (as sum_{i,j} B(i,j) x_i x_j) be a symmetric matrix > with real and positive eigenvalues.

I've found that matrix! It's (2g1 g1 g1 ...) ( g1 2g2 g2 ...) ( g1 g2 2g3 ...) (... ... ... ...) where gi is the square of the logarithm of the (i-1)th prime number. The octave specific equivalent will have a zero eigenvalue. Is that a problem? Here's the general function for intervals. I still don't know how to do more complex wedgies. def intervalComplexity(u, primes=None): """ Geometric complexity for any interval u, expressed as a wedgie """ maxBasis = u.maxBasis() primes = primes or temper.primes[:maxBasis] result = 0.0 for i in range(maxBasis): for j in range(i, maxBasis): result += primes[i]**2 * u[i+1,]*u[j+1,] # back two levels of indentation return math.log(2) * math.sqrt(result)

> Here's the usual grab bag from Weisstein: > > Symmetric Bilinear Form -- from MathWorld * [with cont.]

I'm not sure the point of that.

> http://mathworld.wolfram.com/QuadraticForm.html * [with cont.]

Yes, that looks familiar, and you did mention it in the original message.

> http://mathworld.wolfram.com/PositiveDefiniteQ... * [with cont.]

And that's a special case of the above.

> Hermitian Inner Product -- from MathWorld * [with cont.]

Indeed, I thought it looked more like an inner product than a metric. We can ignore the complex bit, can't we?

> Eh? We are looking at g([0,0], [1, 1]) to find the distance of 15/8 > to 1. If we find the distance of 15/8 to itself, we get zero, which > is correct.

Hey, that's the first time you've mentioned that the metric is from the origin to the point! That's one of the steps that's missing from the original definition.

>>I'm guessing that "note-classes" are something to do with octave >>equivalence -- there's another piece of jargon you haven't defined.

> > I thought "note-classes" was standard music theory jargon for the > equivalence classes defined by octave equivalence.

It's always best to define things that aren't common currency here, in case somebody doesn't know or there's an ambiguity with some other context. I've heard of "pitch classes" which are octave equivalent, but also ignore fine tuning and so work like equal temperaments. I think that makes them "finite cyclic groups". Agmon doesn't seem to call his integer pairs anything. Graham

Message: 7184 Date: Sat, 02 Aug 2003 00:48:11 Subject: Re: Creating a Temperment /Comma From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> In much the same way, I can see that the units depend on the > codimension? of the wedgie. So an interval has complexity measured

in

> octaves, but a linear temperament (or pair of commas) is in

octaves^2.

> Would it make sense to take root so that everything's in octaves?

You could if you wished, I suppose.

>

> > Not correct. All that is required is that the matrix B(i,j) that

you

> > are summing over (as sum_{i,j} B(i,j) x_i x_j) be a symmetric

matrix

> > with real and positive eigenvalues.

> > I've found that matrix! It's > > (2g1 g1 g1 ...) > ( g1 2g2 g2 ...) > ( g1 g2 2g3 ...) > (... ... ... ...) > > where gi is the square of the logarithm of the (i-1)th prime

number. I think I used half of that matrix.

> The octave specific equivalent will have a zero eigenvalue. Is

that a

> problem?

I don't think so--we are ignoring 2. That should make you happy. :)

> Here's the general function for intervals. I still don't know how

to do

> more complex wedgies.

You can either convert to an orthonormal basis, or use my worked-out formulas.

> > Here's the usual grab bag from Weisstein: > > > > Symmetric Bilinear Form -- from MathWorld * [with cont.]

> > I'm not sure the point of that.

This gives the inner product you asked about below.

> > http://mathworld.wolfram.com/QuadraticForm.html * [with cont.]

> > Yes, that looks familiar, and you did mention it in the original

message.

>

> > http://mathworld.wolfram.com/PositiveDefiniteQ... * [with cont.]

> > And that's a special case of the above. >

> > Hermitian Inner Product -- from MathWorld * [with cont.]

> > Indeed, I thought it looked more like an inner product than a

metric.

> We can ignore the complex bit, can't we?

Since everything in sight is real, yes. It makes things that much easier.

> > Eh? We are looking at g([0,0], [1, 1]) to find the distance of

15/8

> > to 1. If we find the distance of 15/8 to itself, we get zero,

which

> > is correct.

> > Hey, that's the first time you've mentioned that the metric is from

the

> origin to the point! That's one of the steps that's missing from

the

> original definition.

That's the norm, or size of the interval, which is where I started from (the L(p/q) function.) You can formulate things in terms of Euclidean metrics, Euclidean norms, inner products, bilinear forms, or quadradic forms, and pass from one to another. It's all closely related.

Message: 7185 Date: Sat, 02 Aug 2003 03:11:52 Subject: Graham definitions for geometric complexity and badness From: Gene Ward Smith I gave a definition of geometric complexity using natural logs; Graham suggested log base 2 instead. He also proposed taking the dth root of this, where d is the codimension of the wedgie--that is, the number of commas used to define it. If G is the Graham geometric complexity under this definition, R is (rms or minimax, etc.) error, and n = pi(p) is the number of primes in the p-limit we are looking at, the formula for geometic badness now becomes B = R G^n which is pretty nice. Does anyone have a concern about switching definitions to the Graham version, which looks to me like a good idea?

Message: 7186 Date: Sat, 2 Aug 2003 01:39:08 Subject: Re: Graham definitions for geometric complexity and badness From: monz@xxxxxxxxx.xxx i suggest we use the name "Breed complexity". feedback please. -monz

> -----Original Message----- > From: Gene Ward Smith [mailto:gwsmith@xxxxx.xxxx > Sent: Friday, August 01, 2003 8:12 PM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] Graham definitions for geometric complexity and > badness > > > I gave a definition of geometric complexity using natural logs; > Graham suggested log base 2 instead. He also proposed taking the dth > root of this, where d is the codimension of the wedgie--that is, the > number of commas used to define it. > > If G is the Graham geometric complexity under this definition, R is > (rms or minimax, etc.) error, and n = pi(p) is the number of primes > in the p-limit we are looking at, the formula for geometic badness > now becomes > > B = R G^n > > which is pretty nice. Does anyone have a concern about switching > definitions to the Graham version, which looks to me like a good idea?

Message: 7187 Date: Sat, 02 Aug 2003 01:39:50 Subject: Re: Graham definitions for geometric complexity and badness From: Carl Lumma

>i suggest we use the name "Breed complexity". >feedback please.

No, no, Graham is a cool first name, and first names should be used when possible. :) -C.

Message: 7188 Date: Sat, 02 Aug 2003 11:45:11 Subject: Re: Graham definitions for geometric complexity and badness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >i suggest we use the name "Breed complexity". > >feedback please.

> > No, no, Graham is a cool first name, and first names > should be used when possible. :)

What's wrong with just calling it geometric complexity? You are going to get this mixed up with Graham's complexity for linear temperaments.

Message: 7189 Date: Sun, 03 Aug 2003 17:43:21 Subject: Re: Creating a Temperment /Comma From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:

> >> >>>consistency is only defined for ets.

> >> >> > >> >>What Gene said about regular temperaments sounded > >> >>equivalent to me.

> >> > > >> >a definition of consistency for regular temperaments? > >> >doesn't seem possible, as they're infinite, so you can > >> >always find better and better approximations to anything.

> >> > >> Yahoo groups: /tuning-math/message/3330 * [with cont.]

> > > >umm . . . so?

> > So, doesn't sound like you can break consistency in > regular temperaments!

how can you break something that isn't even defined?

Message: 7190 Date: Sun, 03 Aug 2003 17:53:54 Subject: Re: review requested From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> --> Whoops, wrong list! > > Hey everybody, > > I wrote this in a hurry, trying to explain the activity on this list > to a friend is as short a document as possible.

i just annotated it with a bunch of corrections, but then i hit the backspace key (which usually acts as "delete") and was sent back to the previous webpage. AAARRRRRGGGGHHHH!!

> I haven't read *The Forms of Tonality* since early 2001, but I plan > to do that again now. I'm sure it covers much of the same ground.

not enough of it. you're getting closer to my whole philosophy on these things.

> I wonder what everyone thinks of this? Anything you disagree with? > Errors? Just not worth fixing?

let me try again, and hopefully it won't all disappear . . .

Message: 7191 Date: Sun, 03 Aug 2003 11:05:54 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>> So, doesn't sound like you can break consistency in >> regular temperaments!

> >how can you break something that isn't even defined?

You can't! Which is why I was said there isn't any in linear temperaments. You mentioned something about infinite numbers of notes always yielding better approximations. What did you mean by that? -Carl

Message: 7192 Date: Sun, 03 Aug 2003 18:06:16 Subject: Re: review requested From: Paul Erlich is the diatonic major second just or near-just?

> () Eventually, we will run into pitches that are very close to

pitches

> we already have. The small intervals between such pairs of pitches > called commas. > > () We create a "pun" if we use the same name ("Ab") for both > notes in such a pair.

i think "pun" refers to the same pitch being used for different musical functions.

> > () We can temper the comma(s) out! > > () Doing so collapses the lattice into a finite "block".

or an infinite "strip", "slice", etc., depending on how many independent commas we temper out.

> The > block tiles the lattice. To move between tiles, simply > transpose all the notes in the basic block by some number of > commas.

if you're tempered the relevant comma out, no transposition is involved at all.

> () You can think of "simple" as giving more intervals > with fewer tones if the comma is tempered out.

??

> > () As a matter of strange coincidence, the same math

is

> behind harmonic entropy!

behind or in front of?

> However, while diminished makes a good showing, you can see that > "porcupine" is better. Note... > > () Diminished, but not porcupine, is to be found in 12-et. > > () Diminished was not used by composers until after 12-et had > become entrenched. > > This suggests that porcupine is a potentially fertile direction for

new

> music. Indeed, the temperament is named after a rather fetching

piece

> by composer Herman Miller, the "Mizarian Porcupine Overture"... > > * [with cont.] (Wayb.) > > Porcupine temperament * [with cont.] (Wayb.)

my piece "glassic" is even more directly based on this temperament, using its 7-note MOS for long stretches, and was just rebroadcast on wnyc!

> > () Reactions... >

> >>() We can temper the comma(s) out! > >> () Doing so collapses the lattice into a finite "block". > >> The block tiles the lattice.

> > > >There you've lost me. I would think that when the lattice

collapsed,

> >that would be the end of it. What is the nature of the lattice, > >once you've tempered the commas out?

> > It collapses to a regular tiling.

or, perhaps better, you could picture it as being wrapped into a cylinder (in the case of the infinite strip), or a torus (in the case of the finite block), etc.

Message: 7193 Date: Sun, 03 Aug 2003 00:33:09 Subject: review requested From: Carl Lumma --> Whoops, wrong list! Hey everybody, I wrote this in a hurry, trying to explain the activity on this list to a friend is as short a document as possible. I haven't read *The Forms of Tonality* since early 2001, but I plan to do that again now. I'm sure it covers much of the same ground. I wonder what everyone thinks of this? Anything you disagree with? Errors? Just not worth fixing? Thanks, -Carl ________________________________________________________________________ () Assume... () Music involves repetition. Sometimes, instead of an exact repetition, a few things are changed while the rest stay the same. () A theme played in a different mode keeps generic intervals (3rds, etc.) the same while pitches change [only true for Rothenberg-proper scales]. () A theme played in a different key keeps absolute intervals the same while pitches change [as goes Rothenberg-efficiency, one relies more on something like the rules of tonal music to make the key-changes recognizeable]. () Possible intervals between notes are to be taken from some fixed set of just or near-just intervals [to exploit the signal- processing capabilities of the hearing system to deliver information to the listener]. () So, to build a scale, we take our chosen just intervals and *stack* them. This generates a lattice. 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. etc. () Eventually, we will run into pitches that are very close to pitches we already have. The small intervals between such pairs of pitches called commas. () We create a "pun" if we use the same name ("Ab") for both notes in such a pair. () 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 temper the comma(s) out! () Doing so collapses the lattice into a finite "block". The block tiles the lattice. To move between tiles, simply transpose all the notes in the basic block by some number of commas. () It appears that most scales that have been popular around the world and throughout history correspond fairly well to temperaments where very "simple" commas have been tempered out. () "Simple" is measured by the distance on the lattice between the two pitches that generate the comma. Thus, simple commas naturally tend to define smaller (fewer pitches) blocks. () You can think of "simple" as giving more intervals with fewer tones if the comma is tempered out. () To further rate temperaments, simple may be balanced against error. The error is determined by the (log) size of the comma and the number of notes in the block over which it must be distributed (which simple already tells us). () That is, it's easy to find small ratios with arbitrarily large denominators (1001:1000). We want temperaments based on the simplest ratios in a given size range. () As a matter of strange coincidence, the same math is behind harmonic entropy! () To see a database of 5-limit temperaments database, go to... Yahoo groups: /tuning/database/ * [with cont.] ...or try the Excel version at... * [with cont.] (Wayb.) ...Try sorting by denominator (the Excel version should be by default). You can see that 81:80 is one of the simplest commas, and is by far the smallest comma among the few most simple (the list itself is the result of searching 5-limit ratio space for low size*denominator ratios). Another comma that looks good is the major diesis (648:625), which leads to the "diminished" temperament. If we take 8 tones/octave of this temperament, we get the octatonic scale of Stravinsky and Messiaen. However, while diminished makes a good showing, you can see that "porcupine" is better. Note... () Diminished, but not porcupine, is to be found in 12-et. () Diminished was not used by composers until after 12-et had become entrenched. This suggests that porcupine is a potentially fertile direction for new music. Indeed, the temperament is named after a rather fetching piece by composer Herman Miller, the "Mizarian Porcupine Overture"... * [with cont.] (Wayb.) Porcupine temperament * [with cont.] (Wayb.) () Reactions...

>>() We can temper the comma(s) out! >> () Doing so collapses the lattice into a finite "block". >> The block tiles the lattice.

> >There you've lost me. I would think that when the lattice collapsed, >that would be the end of it. What is the nature of the lattice, >once you've tempered the commas out?

It collapses to a regular tiling. Here... * [with cont.] (Wayb.) ...the "unison vectors" (Fokker's term for commas) 81:80 and 32:25 define the tile shown in red, which defines the scale shown in green.

>Is it tempered or untempered (infinite or finite)?

Blocks may be left untempered. In the 5-limit, if you temper one comma out you get a linear temperament (chain). Temper both commas out and you get an equal temperament. In the 7-limit, 3 commas are required to tile the lattice. Temper one out and get a planar temperament, two out get a linear temperament, three out get an equal temperament. In the 3-limit, the diatonic scale may be seen as a block defined by the 2187:2048 (apotome), not tempered out. In the 5-limit, it may be defined by 81:80 and 25:24 with the 81:80 tempered out and the 25:24 not tempered out.

>If it's tempered, aren't all instances the same? If it's untempered, >what does it mean to "tile the lattice" with a tempered block?

If all commas are tempered out, you can move from tile to tile without changing any pitches. For some untempered comma, you must transpose all the pitches in your tile by it n times to get the pitches for the nth tile away in that comma's direction (which is why Fokker called it a vector, though Gene says it's not really a vector). -Carl

Message: 7194 Date: Sun, 03 Aug 2003 11:08:03 Subject: Re: review requested From: Carl Lumma

>> I wrote this in a hurry, trying to explain the activity on this list >> to a friend is as short a document as possible.

> >i just annotated it with a bunch of corrections, but then i hit the >backspace key (which usually acts as "delete") and was sent back to >the previous webpage. AAARRRRRGGGGHHHH!!

Sorry that happened. Thanks for trying!

>> I haven't read *The Forms of Tonality* since early 2001, but I plan >> to do that again now. I'm sure it covers much of the same ground.

> >not enough of it. you're getting closer to my whole philosophy on >these things.

I don't think I've changed my position much. What I have is basically from your early posts on this topic. -Carl

Message: 7195 Date: Sun, 03 Aug 2003 11:24:19 Subject: Re: review requested From: Carl Lumma

>is the diatonic major second just or near-just?

Not sure where this is pointed.

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

> >i think "pun" refers to the same pitch being used for different >musical functions.

It was my understanding that it was your assumption that two Ab notes in a score refer to the same pitch, and thus, imply meantone temperament for common practice music. In which case my phrasing above is ok. If we don't want to make that assumption I should change it.

>> () We can temper the comma(s) out! >> >> () Doing so collapses the lattice into a finite "block".

> >or an infinite "strip", "slice", etc., depending on how many >independent commas we temper out. >

>> The >> block tiles the lattice. To move between tiles, simply >> transpose all the notes in the basic block by some number of >> commas.

> >if you're tempered the relevant comma out, no transposition is >involved at all.

Yes, it seems these do not belong as sub-items to tempering it out! Obviously, I'm talking about the untempered case. Huge oversight there.

>> () You can think of "simple" as giving more intervals >> with fewer tones if the comma is tempered out.

> >??

Simple commas tend to define small blocks, so if all commas are tempered out we get all the intervals with fewer notes. Even though a linear temp. has infinitely many notes, there must be something similar going on...

>> () As a matter of strange coincidence, the same math >> is behind harmonic entropy!

> >behind or in front of?

? Anyway, this clearly doesn't belong in the doc. But if you could write a blurb on this for monz or someone to post, I think it'd be interesting.

>> * [with cont.] (Wayb.) >> >> Porcupine temperament * [with cont.] (Wayb.)

> >my piece "glassic" is even more directly based on this temperament, >using its 7-note MOS for long stretches, and was just rebroadcast on >wnyc!

Nice. Do you have a link? If I ever decide to publish this, it will be as a web page with inline graphics, and I'll ask Herman for a link to MPO. Or, I'm happy to provide links at lumma.org.

>>I haven't read *The Forms of Tonality* since early 2001, but I plan >>to do that again now. I'm sure it covers much of the same ground.

> >not enough of it.

I assume you mean I'm not covering enough of your ground? My goal is to make a document much shorter than TFOT. Actually, maybe I haven't even do so. TFOT was pretty short IIRC! -Carl

Message: 7196 Date: Sun, 3 Aug 2003 12:19:18 Subject: Re: review requested From: monz@xxxxxxxxx.xxx hi Carl and paul,

> From: Carl Lumma [mailto:ekin@xxxxx.xxxx > Sent: Sunday, August 03, 2003 11:24 AM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: Re: [tuning-math] Re: review requested > > > <snip> >

> >> () We can temper the comma(s) out! > >> > >> () Doing so collapses the lattice into a finite "block".

> > > >or an infinite "strip", "slice", etc., depending on how many > >independent commas we temper out. > >

> >> The > >> block tiles the lattice. To move between tiles, simply > >> transpose all the notes in the basic block by some number of > >> commas.

> > > >if you're tempered the relevant comma out, no transposition is > >involved at all.

> > Yes, it seems these do not belong as sub-items to tempering it > out! Obviously, I'm talking about the untempered case. Huge > oversight there. >

> >> () You can think of "simple" as giving more intervals > >> with fewer tones if the comma is tempered out.

> > > >??

> > Simple commas tend to define small blocks, so if all commas are > tempered out we get all the intervals with fewer notes. Even though > a linear temp. has infinitely many notes, there must be something > similar going on... > > <snip>

Carl, it's simply a matter of differing dimensions. i think i must be misunderstanding this disucssion, because i'd think that *you* would get this. if i'm having to explain this to you, then i must be missing something. finity does not necessarily imply a dualistic distinction between finite and infinite. assuming that each prime-factor is a unique dimension in a multi-dimensional array characterizing the mathematics of the harmony, as successive commas are tempered out or ignored, there is a subsequent successive reduction of the dimensions by elimination, one dimension at a time for each comma. ... but *you* know that already, no? so then what what are you referring to by "something similar going on"? -monz

Message: 7197 Date: Sun, 03 Aug 2003 21:31:37 Subject: Re: review requested From: Carl Lumma Heya monz,

>>>> () You can think of "simple" as giving more >>>> intervals with fewer tones if the comma is >>>> tempered out.

>>> >>>??

>> >>Simple commas tend to define small blocks, so if all commas are >>tempered out we get all the intervals with fewer notes. Even >>though a linear temp. has infinitely many notes, there must be >>something similar going on...

> >Carl, it's simply a matter of differing dimensions. > >i think i must be misunderstanding this discussion,

I'm not sure which part of the above quote you're referring to. The last part there is a aggressive abstraction of Paul's complexity heuristic, which may not be accurate.

>finity does not necessarily imply a dualistic distinction >between finite and infinite. assuming that each prime-factor >is a unique dimension in a multi-dimensional array >characterizing the mathematics of the harmony, as >successive commas are tempered out or ignored, there is >a subsequent successive reduction of the dimensions >by elimination, one dimension at a time for each comma. > >... but *you* know that already, no? so then what >what are you referring to by "something similar going on"?

That part refers to the simple thing. The simpler the comma, the lower the complexity of the resulting temperament(s). That comes from Paul's heuristic. Complexity can be defined as intervals/notes. -Carl

Message: 7198 Date: Sun, 03 Aug 2003 22:33:21 Subject: Re: Creating a Temperment /Comma From: Carl Lumma

>consistency is only defined when each just interval has a >best approximation in the tuning. with an infinite number >of irrational notes, there is no best approximation, you >can keep finding better and better ones.

But you have to respect the map, according to Gene. So we can still define something like best approx. of n/p must be best approx. n - best approx p. -Carl

Message: 7199 Date: Sun, 03 Aug 2003 22:36:13 Subject: Re: review requested From: Carl Lumma

>>>not enough of it. you're getting closer to my whole philosophy on >>>these things.

>> >> I don't think I've changed my position much.

> >your position? i mean you're getting closer than the forms of >tonality alone.

Oh, thanks. I thought you meant I needed to get closer to it. Then maybe it's worth getting everybody's Seal of Approval on the present doc and publishing it on the web, with inline graphics. I'll need a link to glassic. If you haven't changed the file since the mp3.com days, I have it, and can provide a link for you. -Carl

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