> Given two unison vectors U and V determining a block having periodicity=

> in octave, how to determine the minimal polytope centered around the

> unison in which the transposition space of a possible simplest mode m

> would be contained?

If I start with the basis <16/15,81/80,25/24> I get a corresponding inverse=

matrix [h7,h3,h5]:

[ 7 3 5]

[11 5 8]

[16 7 12]

If I apply your transformation to the two commatic unisons, I get

<16/15,128/135,81/80>^(-1) = [h7,-h5,h3-h5] = [h7,-h5,-h2]

<16/15,25/24,128/135>^(-1) = [h7,h5-h3,-h3] = [h7,h2,-h3]

I can produce corresponding blocks, starting from 1, for each of these by m=

eans of

(16/15)^n (81/80)^round(3n/7) (25/24)^round(5n/7)

(16/15)^n (128/135)^round(-5n/7) (81/80)^round(-2n/7)

(16/15)^n (25/24)^round(2n/7) (128/135)^round(-3n/7)

where n runs from 0 to 6 and "round" means round to the nearest integer.

The scales we get in this way are

scale1: [1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5]

scale2: [1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9]

scale3: [1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8]

There is overlap among these; their union is

[1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 27/16, 16/9, 9/5=

, 15/8];

Does this have anything to do with what you are saying?

> By the way, I would ask if the present state of your maths permits to sho=

w that these

o systems are the simplest in <2 3 5>Z3 ?

I need a definition before I get a statement or a proofΒwhat do you mean by=

"simplest"?

Gene wrote:

The scales we get in this way are

scale1: [1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5]

scale2: [1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9]

scale3: [1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8]

There is overlap among these; their union is

[1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8];

Does this have anything to do with what you are saying?

These scales and their union have certainly to do with what I am saying since there

is confinement in a region near unison and your scales are true modes in the sense of

the gammier theory. Moreover these modes are pretty well chosen to illustrate what

I wrote in my recent posts about unison vectors, hexagonal region, S-matrix, etc.

About the union I will say only that it misses only 27/20 and 40/27 to obtain the

gammier generated by <1 3 5 9 15 27>. But none of the three scales is a sui generis

mode in that gammier, which is the Zarlino gammier <1 3 5 9 15> completed with the

odd 27 and whose steps are <16/15 10/9 9/8>.

I recall here what is a sui generis mode, that I would call "proper" mode if the term

was not already used. On a finite set of intervals modulo 2 with a partial order

defined by the partial composition law in the set, all totally ordered maximal

subset are modes in that structure. Among the sui generis modes of a structure,

those remarkable have a minimal transposition space, what is strongly correlated

to the harmonic properties of such modes.

I recall also that a gammier is obtained like a diamond with the differences between

the elements of an appropriate odd generator which is necessarily non convex.

I will add scale4: <1 9/8 5/4 4/3 3/2 5/3 15/8>, the Zarlino scale, for comparison.

What are the steps of the scales?

scale1: <27/25 10/9 9/8>

scale2: <256/243 9/8>

scale3: <16/15 9/8 75/64>

scale4: <16/15 10/9 9/8>

What are the transposition spaces of these scales, in other words, the intervals spaces

spanned by the elements of these scales, in other words, the gammiers generated by

these scales considered as chordic generators?

scale1: gammier <15 25 27 45 75 81 135>

scale2: gammier <1 3 9 27 81 243>

scale3: gammier <1 3 5 15 45 75 225>

scale4: gammier <1 3 5 9 15 27 45>

What are the S-matrix associated with your scales?

--------------------------------------------------------------------------------

HTML arrays

scale1 27/25 10/9 9/8

27/25 1 250/243 25/24

10/9 243/250 1 81/80

9/8 24/25 80/81 1

scale2 256/243 9/8

256/243 1 2187/2048

9/8 2048/2187 1

scale3 16/15 9/8 75/64

16/15 1 135/128 1125/1024

9/8 128/135 1 25/24

75/64 1024/1125 24/25 1

scale4 16/15 10/9 9/8

16/15 1 25/24 135/128

10/9 24/25 1 81/80

9/8 128/135 80/81 1

--------------------------------------------------------------------------------

What are the hexagones associated with these scales?

scale1: 81/80-25/24-250/243-80/81-24/25-243/250

scale2: no hexagone

scale3: 25/24-1125/1024-135/128-24/25-1024/1125-128/35

scale4: 81/80-135/128-25/24-80/81-128/135-24/25

That corresponds to elements in the matrices in order of a "8" shape.

We can look now at lattice representation in <2 3>Z2/ <2>Z for scale2 and

<2 3 5>Z3 / <2>Z for the others scales and see that the contents of the

segment or the hexagone is precisely the maximal gammier corresponding to

the transposition space. (In blue: class 0 - in red: the scale).

scale1

V . . . . . . 0 . . .

. 2 6 3 0 . . . . . .

. 0 4 1 5 2 . . . . .

. . . 6 3 0 4 1 . . .

. . . . . 5 2 6 3 U .

. . . . . . 0 4 1 5 .

. . . 0 . . . . . . 0

scale2

0 4 1 5 2 6 3 0 4 1 5 2 6 3 U

scale3

. . . . . 0 .

. . V 4 1 5 .

. . 5 2 6 3 U

. 6 3 0 4 1 .

0 4 1 5 2 . .

. 2 6 3 0 . .

. 0 . . . . .

scale4

. . . V . . . . .

0 4 1 5 2 6 3 0 .

. 2 6 3 0 4 1 5 .

. 0 4 1 5 2 6 3 U

. . . . . 0 . . .

What are the periodicity blocks associated with these gammiers in segment or hexagone?

scale1: <81/80 250/243> + <250/243 24/25> + <24/25 81/80>

scale2: <2187/2048> + <2048/2187>

scale3: <135/128 25/24> + <25/24 1024/1125> + <1024/1125 135/128>

scale4: <81/80 25/24> + <25/24 128/135> + <128/135 81/80>

As you may verified, the segment or the hexagone is obtained applying to the first block B

a matrix M being a third root of the identity matrix

[0 -1]

[1 -1]

in 3D and in the 2D case, at the reduced

[-1]

which is the square root of the unity.

--------------------------------------------------------------------------------

Gene wrote:

I need a definition before I get a statement or a proof — what do you mean by "simplest"?

There exist for any "well-structured scale" a minimal gammier in which that scale corresponds

to a sui generis mode. And there exist a canonical order on the gammier space : the order of

their minimal generator.

What are the minimal generators of the minimal gammiers corresponding to our scales?

scale1: <15 25 27 45 135>

scale2: <1 3 27 243>

scale3: <1 5 45 75>

scale4: <1 5 9 15>

For comparison, the three simplest generators in <2 3 5>Z3 are

<1 3 5 9>

<1 3 9 15>

<1 5 9 15>

Pierre

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> What are the transposition spaces of these scales, in other words, the intervals spaces

> spanned by the elements of these scales, in other words, the gammiers generated by

> these scales considered as chordic generators?

> scale1: gammier <15 25 27 45 75 81 135>

> scale2: gammier <1 3 9 27 81 243>

> scale3: gammier <1 3 5 15 45 75 225>

> scale4: gammier <1 3 5 9 15 27 45>

Judging by these examples, I would propose the following definitions:

(1) We may write any positive rational number r in the form

2^n p/q, where p and q are odd integers and p/q is reduced to lowest form. The fraction p/q we call the *odd part* of r.

(2) For any set S of rational numbers, we may take the odd parts of each element, and the least common multiple D of their denominators. The set of integers gammier(S) is defined as the set of the odd parts of the elements of S times D; gammier(S) is the *gammier of S*.

Is there anything about this you want to accept, or to modify and then accept?

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Sunday, December 23, 2001 3:25 AM

> Subject: [tuning-math] Re: For Pierre, from tuning

>

>

> --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

>

> > What are the transposition spaces of these scales, in other words, the

intervals spaces

> > spanned by the elements of these scales, in other words, the gammiers

generated by

> > these scales considered as chordic generators?

> > scale1: gammier <15 25 27 45 75 81 135>

> > scale2: gammier <1 3 9 27 81 243>

> > scale3: gammier <1 3 5 15 45 75 225>

> > scale4: gammier <1 3 5 9 15 27 45>

>

> Judging by these examples, I would propose the following definitions:

>

> (1) We may write any positive rational number r in the form

> 2^n p/q, where p and q are odd integers and p/q is reduced

> to lowest form. The fraction p/q we call the *odd part* of r.

I like this.

What's interesting to me is to ponder the difference between

thinking of a quantity in this form as opposed to the one

I've preferred, which is simply 2^x 3^y 5^z... P^n, where

P is the limiting prime-factor and x,y,z,n are integers

(or often lately, fractions of integer terms).

So is the general consensus that the former [2^n p/q] is

best for describing dyads/intervals, and the latter

[2^x 3^y 5^z... P^n] is best for describing larger entities

such as entire tuning systems?

What about those cases falling between, such as tri-, tetr-,

pent-, hex-ads etc.? Most of you feel that the 2^n p/q

notation is best for these, yes?

-monz

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

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

> So is the general consensus that the former [2^n p/q] is

> best for describing dyads/intervals,

Yes, if octave-equivalence is assumed.

> and the latter

> [2^x 3^y 5^z... P^n] is best for describing larger entities

> such as entire tuning systems?

This doesn't make sense, as you'll only be describing a _single

ratio_ here, and that single ratio can not correspond to more notes

than a dyad.

However, I would agree that if one is looking at JI tuning systems,

the highest prime number P is one of the most important pieces of

information you could want about a system . . . as well as which, if

any, primes are not used.

Gene wrote:

Judging by these examples, I would propose the following definitions:

(1) We may write any positive rational number r in the form

2^n p/q, where p and q are odd integers and p/q is reduced to lowest form. The

fraction p/q we call the *odd part* of r.

(2) For any set S of rational numbers, we may take the odd parts of each

element, and the least common multiple D of their denominators. The set of

integers gammier(S) is defined as the set of the odd parts of the elements of S

times D; gammier(S) is the *gammier of S*.

Is there anything about this you want to accept, or to modify and then accept?

There is no problem with (1).

I would add only the terms I use for that.

r = 2n p/q is the rational number

r mod 2 = {2x p/q | all x in Z} is the corresponding octave class

Ton ( r ) = 2k p/q | k such that r in [1,2[ is the tone representing the class

Pivot ( r ) = p/q is the pivot representing the class : your odd part.

There are few problems with (2).

Gammier and generator

First, the expression "gammier <a b c ...>" means the structure generated by

<a b c ...>. The term <a b c ...> is refered as the generator g of the structure,

while the elements of that structure g\g are

a\a a\b a\c ...

b\a b\b b\c ,,,

c\a c\b c\c ...

,,,

where u\v means here the interval between u and v. The symbol \ is independant

of the composition law type, multiplicative or additive. So

a.. 4\5 = 5/4

b.. (log4)\(log5) = log5 - log4

Gammier conditions

More important, the gammier structure implies the existence of four conditions,

the last axioms of the gammier structure, which are

a.. regularity

b.. contiguity

c.. congruity

d.. fertility

so you refer simply to the harmoid structure if these conditions being unknow,

you use only

a.. this type of generation derived from the chordoid theory

b.. giving a finite set of rational numbers (mod 2)

implying implicitely

a.. the appropriate restriction of the multiplication as the law

b.. and the standard rational ordering

all that being necessary to formulate the axioms.

Minimal odd generator

Any finite set of rational numbers may be considered as a chordic generator of an

harmoid. Any line and any column of the matrix g\g, generated by a such chordic

generator g, may generate the same harmoid. There exist also an infinity of odd

set <a b c ...> generating the same harmoid.

It is important to find the minimal odd generator of a given harmoid. The canonical

order on the space of harmoids corresponds to the order of their minimal generator.

Your definition may permit to find it but also may fail to find that minimal generator.

For instance,

<1 10/9 5/4 5/3 20/11>

is a chordic generator of the gammier number 4 which contains the rast scale. Your

definition gives the lcm D = 99 and then the following odd generator

<99 110 165 180 495>

while the minimal generator is

<1 3 5 9 11>

I imagine you have already understood the problem linked to the duality in chordoid

structures: lines and columns of the chordic matrices being equigenerative.

There exist two ways to reduce a set of "odd parts" and we have to compare them

to find the minimal odd set.

Pierre.

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> r mod 2 = {2^x p/q | all x in Z} is the corresponding octave class

There's a problem with this--number theorists already mean something quite specific by r mod 2, and this isn't it. r mod 2 is 0 if x>0,

is 1 if x=0, and is 'infinity' or undefined if x<0. How about

<2^n r> for this?

The term <a b c ...> is refered as the generator g of the structure,

> while the elements of that structure g\g are

> a\a a\b a\c ...

> b\a b\b b\c ,,,

> c\a c\b c\c ...

> ,,,

> where u\v means here the interval between u and v. The symbol \ is independant

> of the composition law type, multiplicative or additive. So

> a.. 4\5 = 5/4

> b.. (log4)\(log5) = log5 - log4

So if {a, b, c} is a set of odd integers, the structure it generates are the ratios greater than one between them?

> Gammier conditions

> More important, the gammier structure implies the existence of four conditions,

> the last axioms of the gammier structure, which are

> a.. regularity

> b.. contiguity

> c.. congruity

> d.. fertility

> so you refer simply to the harmoid structure if these conditions being unknow,

> you use only

> a.. this type of generation derived from the chordoid theory

> b.. giving a finite set of rational numbers (mod 2)

> implying implicitely

> a.. the appropriate restriction of the multiplication as the law

> b.. and the standard rational ordering

> all that being necessary to formulate the axioms.

>

> Minimal odd generator

> Any finite set of rational numbers may be considered as a chordic generator of an

> harmoid. Any line and any column of the matrix g\g, generated by a such chordic

> generator g, may generate the same harmoid. There exist also an infinity of odd

> set <a b c ...> generating the same harmoid.

>

> It is important to find the minimal odd generator of a given harmoid. The canonical

> order on the space of harmoids corresponds to the order of their minimal generator.

>

> Your definition may permit to find it but also may fail to find that minimal generator.

> For instance,

> <1 10/9 5/4 5/3 20/11>

> is a chordic generator of the gammier number 4 which contains the rast scale. Your

> definition gives the lcm D = 99 and then the following odd generator

> <99 110 165 180 495>

I get <396,440,495,660,720>, with the ratios they generate being

1, 12/11, 10/9, 9/8, 5/4, 4/3, 16/11, 3/2, 18/11, 5/3, 20/11

However, taking 2 out of the picture gives me

<1, 5/9, 5, 5/3, 5/11>

which leads to

<45,55,99,165,495> and finally to

1,3,5,9,5/3,9/5,11/3,11/5,11/9

which is not at all the same.

> while the minimal generator is

> <1 3 5 9 11>

From this I get

1,3,5,9,5/3,9/5,11/3,11/5,11/9, with an extra 11 in there.

From this I get

> I imagine you have already understood the problem linked to the duality in chordoid

> structures: lines and columns of the chordic matrices being equigenerative.

I don't understand the above sentence; to follow you, I need clear, mathematical definitions.

Gene wote:

> r mod 2 = {2^x p/q | all x in Z} is the corresponding octave class

There's a problem with this--number theorists already mean something quite

specific by r mod 2, and this isn't it. r mod 2 is 0 if x>0,

is 1 if x=0, and is 'infinity' or undefined if x<0. How about

<2^n r> for this?

I made an inattention error here. Forget r mod 2 = in the line and read simply

{2^x p/q | all x in Z} is the corresponding octave class

I use normally r mod <2> as equivalent to Ton (r) or Pivot (r) implying only

a multiplicative modulo.

--------------------------------------------------------------------------------

Gene wrote:

So if {a, b, c} is a set of odd integers, the structure it generates are the

ratios greater than one between them?

The structure has two levels. I don't have time to explain that in details. I will use

simply an example.

Let {a, b, c} = {1, 3, 5}. The ordered odd generator g is noted <1 5 3> and the

corresponding harmoid structure is represented by the chordic matrix g\g

.1. 5/4 3/2

8/5 .1. 6/5

4/3 5/3 .1.

The ordered contents '(g\g)

{1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}

is the class level of the corresponding harmoid structure. I use most often the

tones of the first octave to represent the classes. In that case the ratios are

greater than one. But there exist equivalent manners to represent the classes:

the pivots and also the tones of the centered octave and then the ratios are

not all greater than one.

{1/5, 1/3, 3/5, 1, 5/3, 3/1, 5/1}

{2/3, 4/5, 5/6, 1, 6/5, 5/4, 3/2}

The harmoid structure is not only a class structure, it's first an interval structure:

{... 3/10, 3/5, 6/5, 12/5, ...}

for instance, belong to the structure at interval level.

I add here that the three first axioms are respected: that harmoid is a gammoid.

--------------------------------------------------------------------------------

> <99 110 165 180 495>

Opps! I forgot to reduce. So <99 55 165 45 495>.

--------------------------------------------------------------------------------

Gene wrote:

However, taking 2 out of the picture gives me

<1, 5/9, 5, 5/3, 5/11>

which leads to

<45,55,99,165,495> and finally to

1,3,5,9,5/3,9/5,11/3,11/5,11/9

which is not at all the same.

> while the minimal generator is

> <1 3 5 9 11>

From this I get

1,3,5,9,5/3,9/5,11/3,11/5,11/9, with an extra 11 in there.

Surely an error here.

495/45 = 11/1

Ton(11/1) = 11/8

--------------------------------------------------------------------------------

Gene wrote:

> I imagine you have already understood the problem linked to the duality in chordoid

> structures: lines and columns of the chordic matrices being equigenerative.

I don't understand the above sentence; to follow you, I need clear,

mathematical definitions.

Ok, forget that for the moment. I thought you would have seen that there exist two

ways to reduce a set of rational numbers to a set of odds. You have used only one. Using

the precedent example, reduced first to pivots, I hope it will be now clear. These sets

are strictly identical:

{1, 5/9, 5/1, 5/3, 5/11} =

{99/99, 55/99, 495/99, 165/99, 45/99} =

{5/5, 5/9, 5/1, 5/3, 5/11}

--------------------------------------------------------------------------------

I close here that session. I could only reply very shortly in these days.

Merry Christmas!

Pierre

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> Let {a, b, c} = {1, 3, 5}. The ordered odd generator g is noted <1 5 3> and the

> corresponding harmoid structure is represented by the chordic matrix g\g

> .1. 5/4 3/2

> 8/5 .1. 6/5

> 4/3 5/3 .1.

This is Partch's "Tonality Diamond", but what does the order do for you? Partch called it arbitary, and it seems to me that he got that right.

In any case, "tonality diamond" is the recognized name here.

> The ordered contents '(g\g)

> {1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}

> is the class level of the corresponding harmoid structure.

I use most often the

> tones of the first octave to represent the classes. In that case the ratios are

> greater than one. But there exist equivalent manners to represent the classes:

Let's see then--the "structure" in your nomenclature of a set of odd integers is a set of octave equivalence class representatives of the elements of the corresponding tonality diamond, ordered by size?

> The harmoid structure is not only a class structure, it's first an interval structure:

> {... 3/10, 3/5, 6/5, 12/5, ...}

> for instance, belong to the structure at interval level.

You've got to give different names to different things.

> I add here that the three first axioms are respected: that harmoid is a gammoid.

This does not convey much without precise definitions.

Gene wrote:

This is Partch's "Tonality Diamond", but what does the order do for you? Partch

called it arbitary, and it seems to me that he got that right.

In any case, "tonality diamond" is the recognized name here.

Shortly.

May I conclude you didn't know that the Zarlino gammier corresponds to a

matrix like the "Tonality Diamond"?

I repeated often that the chordic matrix is like the "Tonality Diamond", but

this matrix is justified by the chordoid theory, which reconstruct the abelian

group ending, rather than starting, with the closure axiom. So, without the

closure, that structure has well-defined properties like being generated by

a chordic matrix.

It's not only limited to convex generators of N-limit type. More, none of the

N-limit diamond is a gammier, the non-convexity is essential to respect the

fertility condition. Many are even not gammoid, like the 11-limit structure,

having not the CS property.

The order has no importance. The choice of an order is only to permit direct

reading, fo instance, of

a.. the treillis which exhibit modes, like <15 1 9 5>

b.. the chords structure, like <1 5 3 15 9>

I will give later the axioms definition with their sense. Now it's holidays.

Pierre

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> Gene wrote:

> This is Partch's "Tonality Diamond", but what does the order do for you? Partch

> called it arbitary, and it seems to me that he got that right.

>

> In any case, "tonality diamond" is the recognized name here.

> Shortly.

>

> May I conclude you didn't know that the Zarlino gammier corresponds to a

> matrix like the "Tonality Diamond"?

Since the Zarlino gammier was not mentioned, I don't think you can conclude anything. However, I am the last person to suspect of having deep knowledge of the history of tuning theory, and to start with I need to ask if by the Zarlino gammier you mean the JI diatonic scale?

> I repeated often that the chordic matrix is like the "Tonality Diamond", but

> this matrix is justified by the chordoid theory, which reconstruct the abelian

> group ending, rather than starting, with the closure axiom.

It seems to me that the chordic matrix *is* the tonality diamond. Wherein do they differ? I can't see any abelian group structure in either, myself, because of the *lack* of closure.

> It's not only limited to convex generators of N-limit type. More, none of the

> N-limit diamond is a gammier, the non-convexity is essential to respect the

> fertility condition. Many are even not gammoid, like the 11-limit structure,

> having not the CS property.

Again, without precise definitions mathematics can't operate. You need to clue me in.

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

> --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

>

> > Gene wrote:

> > This is Partch's "Tonality Diamond", but what does the order do

for you? Partch

> > called it arbitary, and it seems to me that he got that right.

> >

> > In any case, "tonality diamond" is the recognized name here.

> > Shortly.

> >

> > May I conclude you didn't know that the Zarlino gammier

corresponds to a

> > matrix like the "Tonality Diamond"?

>

> Since the Zarlino gammier was not mentioned, I don't think you can

>conclude anything. However, I am the last person to suspect of

>having deep knowledge of the history of tuning theory,

Don't worry -- the history of tuning theory will provide no clue as

to what "Zarlino gammier" means.

>and to start with I need to ask if by the Zarlino gammier you mean

>the JI diatonic scale?

He means the set of 19 ratios "mod 2" representing all the intervals

that are found between pitches in the JI diatonic scale.

Gene,

I have really no time to define all with precision on that forum. If you want more

precision, you could find someone to translate that: the first definitions permitting

to explicit my chord theorem which suggests the chordicity as next axiom, just

before the closure giving then the abelian group.

--------------------------------------------------------------------------------

Structure de simploïde

Soit E un ensemble. Une partie non-vide S de E x E x E est un simploïde sur E, si elle admet les relations ternaires

(x,y,z) = (x,y,z')

(x,y,z) = (x,y',z)

La première relation détermine une loi de composition partielle dans E

pr12<S> --> pr3<S>

qui fait de S un groupoïde large et, réciproquement, la seconde relation détermine à son tour une loi d'accordance partielle dans E

\ : pr13<S> --> pr2<S>

Comme ces lois se déterminent mutuellement par la bijection canonique

((x,y),z) <--> ((x,z),y)

la donnée (parfois possible), pour un simploïde <E,S>, d'un graphe valué (N,F,V), appelé graphe d'accordance, où

les noeuds N = E

les flèches F = pr13<S>

les valeurs V = \(F)

équivaut strictement à la donnée de la table de composition du simploïde, dont les entrées, lorsqu'elles ne sont pas vides, correspondent, pour chaque flèche, à T(source, valeur) = but.

On peut montrer aisément que l'ajout de la seconde relation correspond à celui de l'axiome de simplicité à droite défini comme

ak = ak' implique k = k'

Commutativité et associativité

La commutativité et l'associativité sont généralement définies, en dépendance de la fermeture algébrique,

ab = ba

a(bc) = (ab)c

De façon autonome, sans requérir la fermeture, on peut redéfinir ainsi les axiomes de commutativité et d'associativité, où cette dernière est réduite à une associativité à droite

k = ab implique k = ba

ak = (ab)c implique k = bc

Ajouter la commutativité à un simploïde entraîne la simplicité à gauche, ou encore, pour compléter les deux relations ternaires initiales, la relation

(x,y,z) = (x',y,z)

De même, en ajoutant la commutativité, l'associativité à droite entraîne l'associativité à gauche et on peut ainsi parler simplement de simploïde commutatif et associatif.

Lemme de composition transitive

L'existence des intervalles a\b, b\c et a\c dans un simploïde muni de l'associativité à droite implique la composition transitive

(a\b)(b\c) = a\c

Démonstration.

Dans l'axiome d'associativité à droite

ak = (ab)c implique k = bc

la partie gauche de l'implication

ak = (ab)c

s'explicite

(a,b,x) appartient à S // où x = ab

(x,c,y) appartient à S // où y = xc = (ab)c

(a,k,z) appartient à S // où z = ak = (ab)c = y

et la partie droite k = bc s'explicite

(b,c,k) appartient à S // où k = bc.

En utilisant les trois intervalles de la partie gauche, et l'égalité z = y, la partie droite peut être réécrite

(a\x,x\y,a\y) appartient à S

ce qui se réécrit encore, en assumant l'existence des intervalles

(a\x)(x\y) = a\y

Intervalles, accords, faisceaux

Dans un simploïde, les équations linéaires de la forme ax = b ont, tout au plus, une solution. Cette solution, lorqu'elle existe, est notée a\b et appelée intervalle entre a et b. On peut dire autrement que a est accordé à b.

La loi d'accordance \ peut s'étendre à l'ensemble des parties p(E). Soit A et B deux éléments de p(E). Si pour tout élément (a,b) de A x B il existe un élément k dans E tel que ak = b, on peut dès lors dire que A est accordé à B, et que l'équation AX = B a une solution K, notée A\B, qui désigne l'élément de p(E) correspondant aux intervalles sous-tendus par (A,B).

Définition : Une partie A de E accordée à elle-même, est appelée un accord dans E.

L'élément A\A de p(E), la solution de AX = A, formé des intervalles sous-tendus par l'accordance interne de A, est dit le domaine généré par A, lequel est dit un générateur chordique de ce domaine.

Deux accords A et B dans un simploïde sont

a.. égaux s'ils ont les mêmes éléments

b.. équigénératifs s'ils sous-tendent le même domaine A\A = B\B

c.. équipollents s'il existe une bijection f : A --> B tel que pour tout x et y dans A

f(x)\f(y) = x\y

Soit k un élément de l'accord A. Les équations {k}X = A et AX = {k} ont forcément une solution dans p(E). Ce sont des parties du domaine A\A appelées respectivement faisceau divergent k\A et faisceau convergent A\k du domaine A\A.

Par convention, l'absence de parenthèses est réservée à la notation des faisceaux. Pour pouvoir écrire k\A et A\k au lieu de {k}\A et A\{k}, qui sont des éléments de p(E), et où {k} est un ensemble à un élément, il faut que A soit un accord et que k soit un élément de cet accord.

Théorème des accords

Dans un simploïde,

a.. les faisceaux divergents k\A sont des accords équipollents à A, s'il est muni de l'associativité à droite ;

b.. les faisceaux convergents A\k sont des accords équigénératifs à A, et équipollents entre eux, s'il est muni en plus de la commutativité.

Démonstration.

Soit <E,S> un simploïde sur E muni de l'associativité à droite, et k un élément de l'accord A dans ce simploïde.

Deux éléments génériques x et y de A et l'élément k étant toujours accordés entre eux, le lemme de composition transitive, qui découle de l'associativité à droite, permet d'écrire la relation toujours avérée

(k\x)(x\y) = k\y

impliquant que k\x est toujours accordé à k\y

(k\x)\(k\y) = (x\y)

et, par extension à A tout entier, que le faisceau k\A est bien accordé à lui-même, et constitue, de ce fait, un accord dans <E,S>.

Soit f une application de l'accord A sur l'accord k\A telle que x --> k\x. Puisque la relation précédente peut se réécrire

f(x)\f(y) = x\y

il ne reste à montrer, pour assurer que k\A est bien équipollent à A, que f est bijective, autrement dit que f(x) = f(y) entraîne x = y.

De la relation toujours avérée k(k\x) = x on tire

k f(x) = x

qui permet d'expliciter ce lien

f(x) = f(y)

k f(x) = k f(y)

x = y

(...)

--------------------------------------------------------------------------------

I cut here, after the demonstration of the first part of the chord theorem.

Pierre

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> I have really no time to define all with precision on that forum. If you want more

>

> precision, you could find someone to translate that: the first definitions permitting

>

> to explicit my chord theorem which suggests the chordicity as next axiom, just

>

> before the closure giving then the abelian group.

After struggling through this, I still don't know why you want to mess around with groupoids. Why not simply go to the abelian group, and stay there? You present some definitions, but they seem unmotivated, in other words.

G. Why not simply go to the abelian group, and stay there?

P. I don't go to the abelian group for closure is not possible (without temperament)

in finite JI group. How determine a pertinent region in an infinite JI group?

G. You present some definitions, but they seem unmotivated, in other words.

P. It's your point of view. It's really funny. When I see the orgy of numbers and

technical talk here without any justification, I could ask me if all that is motivated.

Peace on earth at men of good will!

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> I have really no time to define all with precision on that forum. If you want more

>

> precision, you could find someone to translate that: the first definitions permitting

>

> to explicit my chord theorem which suggests the chordicity as next axiom, just

>

> before the closure giving then the abelian group.

After struggling through this, I still don't know why you want to mess around with groupoids. Why not simply go to the abelian group, and stay there? You present some definitions, but they seem unmotivated, in other words.

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> P. I don't go to the abelian group for closure is not possible (without temperament)

> in finite JI group. How determine a pertinent region in an infinite JI group?

This doesn't strike me as a very good reason--why not work within the group, and define whatever regions or limitations you need? Ordinarily you would not limit yourself to a groupoid when there is a group available; in fact the opposite is more commonly seen--when a group is not immediately available, we construct it. The point is always to make things as easy and elegant as possible.

Gene wrote:

This doesn't strike me as a very good reason--why not work within the group,

and define whatever regions or limitations you need? Ordinarily you would not

limit yourself to a groupoid when there is a group available; in fact the

opposite is more commonly seen--when a group is not immediately available, we

construct it. The point is always to make things as easy and elegant as

possible.

Gene,

It's not my style to define arbitrarily limitations I would need.

Since centuries all searchers don't face the conflict between justness of chords

and closure: they seek only the compromise of good temperaments. I choose

another way. It may seem, at your viewpoint, less elegant to find the founding

axioms of the paradigmatic operative structures in music. I don't think so.

I would like to add I don't work with groupoid since groupoid has only one axiom

which is precisely the closure axiom. I constructed the chordoid structure

which have not the closure axiom, but all others axioms of the abelian group.

Could you construct a finite JI group using independant primes?

Pierre.

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> Could you construct a finite JI group using independant primes?

Every element of a finite group has a finite order. However, why would I want to look at a finite JI set as an algebraic object, unless I was going to use the morphisms of the corresponding category somehow? Is this what you do? I'm trying to find out how you think this point of view helps.

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> Gene wrote:

> This doesn't strike me as a very good reason--why not work within

the group,

> and define whatever regions or limitations you need? Ordinarily

you would not

> limit yourself to a groupoid when there is a group available; in

fact the

> opposite is more commonly seen--when a group is not immediately

available, we

> construct it. The point is always to make things as easy and

elegant as

> possible.

> Gene,

>

> It's not my style to define arbitrarily limitations I would need.

>

> Since centuries all searchers don't face the conflict between

justness of chords

> and closure: they seek only the compromise of good temperaments. I

choose

> another way.

Sounds like a motivation for periodicity blocks; I wish to understand

how it motivates you in yet another direction.

> It may seem, at your viewpoint, less elegant to find the founding

> axioms of the paradigmatic operative structures in music.

I seriously doubt Gene would say that. In fact, I bet he could lay

out an axiomatic system for the researches we engage in most of the

time on this list lately.

>

> Could you construct a finite JI group using independant primes?

This looks _exactly_ like what we and especially Gene have been doing

here on this list.

Using a set of unison vectors, you define a periodicity block.

Now, by treating each unison vector as an equivalence relation

(choice of either chromatic or commatic equivalence), you get a

finite group, constructed using independent primes.

If there are no commatic equivalences you wish to temper out, you're

done -- you have a JI block (whose precise ratios can be chosen in a

variety of ways, corresponding to different choices for the shape

(normally convex) and the position of the PB shape, subject to the

constraint that the shape tile the plane with the right symmetry

group).

We've been doing all these sorts of things all along, and yet you

claim to understand nothing. So what are you adding to our

understanding? So far, all I can see in your work is lots of pretty

numbers. How odd!

A very merry christmas to you, my friend, Pierre!

-Paul

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

> --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

>

> > Could you construct a finite JI group using independant primes?

>

> Every element of a finite group has a finite order. However, why

>would I want to look at a finite JI set as an algebraic object,

>unless I was going to use the morphisms of the corresponding

>category somehow? Is this what you do? I'm trying to find out how

>you think this point of view helps.

Gene, if you think you're speaking Pierre's language, is there any

way you might be able to try and explain what we're (you're) doing,

in _his_ language? He said it looks like nothing but numbers to him.

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

> Gene, if you think you're speaking Pierre's language, is there any

> way you might be able to try and explain what we're (you're) doing,

> in _his_ language? He said it looks like nothing but numbers to him.

For Pierre:

The p-limit positive rationals 2^e1 ... p^ek form a free group of rank k under multiplication. An equal temperament can be viewed as an epimorphism of this group to Z. This can be defined in two ways--by giving the map (the equal temperament view) or giving the kernel in terms of a set of generators for the kernel. Then we have the quesiton of the tuning of this system, which involves an injective mapping into the reals. Normally, we tune by making octaves pure.

We've been looking at temperaments, which generalize the above to maps from the group of rank k to a free group of rank 1<m<k; here we traditionally refer to the dimension as m-1, so that rank 2 is a linear temperament, rank 3 planar, and so forth. Just as in the equal temperament case, we have two ways of defining the temperament, by means of a mapping or by defining the kernel, and we also have the question of the tuning injection of the temperament, and the optimal choices for the precise values involved; this is an optimization problem, which suggests using least squares or linear programming for a solution. The question of precisely how to define the goodness of a temperament has much occupied us of late, and invovles Diophantine approximation theory, just as it does in the case of equal temperaments.

In order to survey temperaments, we need to have a standard invariant form to reduce them to. One possibility is to use lattice basis reduction on the kernel, and another is to construct a standard form of the mapping; however I've discovered that invoking multilinear algebra in the form of the wedge product allows one to define an invariant, starting either from mappings or kernels, which uniquely defines the resulting temperament. We've been using this to survey temperaments.

We've also devoted quite a lot of time to JI scales, mostly of those (Fokker blocks and the like) which are preimages of an equal temperament mapping.

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

> For Pierre:

>

> The p-limit positive rationals 2^e1 ... p^ek form a free group of

>rank k under multiplication. An equal temperament can be viewed as

>an epimorphism of this group to Z. This can be defined in two ways--

>by giving the map (the equal temperament view) or giving the kernel

>in terms of a set of generators for the kernel.

Which we call unison vectors.

> We've also devoted quite a lot of time to JI scales, mostly of

>those (Fokker blocks and the like) which are preimages of an equal

>temperament mapping.

Or of a mapping to a partially tempered system, where the unison

vectors that are not tempered out are called "chromatic unison

vectors" -- the 81:80 tempered, and 25:24 (or 135:128) chromatic,

case, correctly accounts for 99% of Western pitch usage since 1480 --

including Zarlino!

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

> We've also devoted quite a lot of time to JI scales, mostly of those (Fokker blocks and the like) which are preimages of an equal temperament mapping.

It occurs to me that this property, which is very important, hasn't been singled out or named so far as I know. I propose to call it the "epimorphic property". For instance, let's see if Margo's

pelog-pentatonic is epimorphic.

The scale is in 2^i 3^j 7^k, so we can leave 5 out of the map. If we denote it by h, and if h(2)=a, h(3)=b and h(7)=c, we want

h(28/27)=1, h(4/3)=2, h(3/2)=3. Solving the resulting linear equations gives a=5, b=8, c=15, and so h(14/9)=4. The scale therefore is epimorphic, or has the epimorphic property.

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

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

>

> > We've also devoted quite a lot of time to JI scales, mostly of

>>those (Fokker blocks and the like) which are preimages of an equal

>>temperament mapping.

>

> It occurs to me that this property, which is very important, hasn't

> been singled out or named so far as I know.

You're kidding? Isn't this equivalent to the PB property or the CS

property for JI scales?

> I propose to call it the "epimorphic property". For instance, let's

>see if Margo's

> pelog-pentatonic is epimorphic.

>

> The scale is in 2^i 3^j 7^k, so we can leave 5 out of the map. If

>we denote it by h, and if h(2)=a, h(3)=b and h(7)=c, we want

> h(28/27)=1, h(4/3)=2, h(3/2)=3. Solving the resulting linear

>equations gives a=5, b=8, c=15, and so h(14/9)=4. The scale

>therefore is epimorphic, or has the epimorphic property.

It's a PB. It's CS. What's new?

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

> You're kidding? Isn't this equivalent to the PB property or the CS

> property for JI scales?

PB ==> epimorphic ==> CS but not conversely, if I've got the definitions right.

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

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

>

> > You're kidding? Isn't this equivalent to the PB property or the

CS

> > property for JI scales?

>

> PB ==> epimorphic ==> CS but not conversely, if I've got the >

definitions right.

OK, but CS ==> PB in all "reasonable" cases where the unison vectors

are not "ridiculously large" relative to the step sizes -- right?

(Clearly a definition of "ridiculously large" is needed.)

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

> OK, but CS ==> PB in all "reasonable" cases where the unison vectors

> are not "ridiculously large" relative to the step sizes -- right?

> (Clearly a definition of "ridiculously large" is needed.)

How does CS allow you to conclude you even have unison vectors? Having enough unison vectors to define the map is equivalent to being epimorphic, by the way.

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

> How does CS allow you to conclude you even have unison vectors? Having enough unison vectors to define the map is equivalent to being epimorphic, by the way.

Plus, the map has to correctly order the scale, so we do need a little more.

> From: paulerlich <paul@stretch-music.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Monday, December 24, 2001 10:29 PM

> Subject: [tuning-math] Re: For Pierre, from tuning

>

>

> --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

>

> > It's not my style to define arbitrarily limitations I would need.

> >

> > Since centuries all searchers don't face the conflict between

> > justness of chords and closure: they seek only the compromise

> > of good temperaments. I choose another way.

>

> Sounds like a motivation for periodicity blocks; I wish to understand

> how it motivates you in yet another direction.

That was exactly my first thought when I read this.

-monz

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Monday, December 24, 2001 11:01 PM

> Subject: [tuning-math] Re: For Pierre, from tuning

>

>

> We've also devoted quite a lot of time to JI scales,

> mostly of those (Fokker blocks and the like) which are

> preimages of an equal temperament mapping.

Hmmm... this description sounds very much like what I'm

trying to portray with my "acoustical rational implications

of meantones" lattices. The JI periodicity-blocks I derive

could be called "preimages of a meantone mapping", which in

turn in many cases equate to an equal-temperament mapping.

-monz

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

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

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

>

> > OK, but CS ==> PB in all "reasonable" cases where the unison

vectors

> > are not "ridiculously large" relative to the step sizes -- right?

> > (Clearly a definition of "ridiculously large" is needed.)

>

> How does CS allow you to conclude you even have unison vectors?

I've never seen a counterexample.

>Having enough unison vectors to define the map is equivalent to

>being epimorphic, by the way.

So can you come up with a counterexample?

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

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

>

> > How does CS allow you to conclude you even have unison vectors?

>Having enough unison vectors to define the map is equivalent to

>being epimorphic, by the way.

>

> Plus, the map has to correctly order the scale, so we do need a

>little more.

Example, please.

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

> OK, but CS ==> PB in all "reasonable" cases where the unison vectors

> are not "ridiculously large" relative to the step sizes -- right?

> (Clearly a definition of "ridiculously large" is needed.)

Presumably, however you define it, PB requires convexity, in which case

1--9/8--5/4--4/3--40/27--5/3--15/8

would be an example of a scale which was epimorphic but not PB. Similarly, assuming (always a possibility) I understand Joe's definition correctly,

1--9/8--5/4--4/3--1024/675--5/3--15/8

would be an example of a scale which is CS, but neither epimorphic nor PB.

How am I doing?

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

>

> > From: genewardsmith <genewardsmith@j...>

> > To: <tuning-math@y...>

> > Sent: Monday, December 24, 2001 11:01 PM

> > Subject: [tuning-math] Re: For Pierre, from tuning

> >

> >

> > We've also devoted quite a lot of time to JI scales,

> > mostly of those (Fokker blocks and the like) which are

> > preimages of an equal temperament mapping.

>

> Hmmm... this description sounds very much like what I'm

> trying to portray with my "acoustical rational implications

> of meantones" lattices. The JI periodicity-blocks I derive

> could be called "preimages of a meantone mapping", which in

> turn in many cases equate to an equal-temperament mapping.

Right . . . but you seem too concerned with certain qualities of the

preimage which end up having no relevance once the meantone and/or ET

tempering happens. For example all the 55-tone periodicity blocks I

gave are exactly equivalent to one another once the 81:80 is tempered

out -- so there's no point in preferring one to another, unless you

really intend to use the JI tuning, wolves and all, rather than the

meantone temperament.

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

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

>

> > OK, but CS ==> PB in all "reasonable" cases where the unison

vectors

> > are not "ridiculously large" relative to the step sizes -- right?

> > (Clearly a definition of "ridiculously large" is needed.)

>

> Presumably, however you define it, PB requires convexity,

I just said "any shape that tiles the plane". Certainly, I've also

tended to impose convexity on top of the PB property whenever a JI,

untempered scale is meant.

Do any of your examples still apply?

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

> I just said "any shape that tiles the plane". Certainly, I've also

> tended to impose convexity on top of the PB property whenever a JI,

> untempered scale is meant.

Does "shape" entail connectedness, or can it be scattered islands all over the place?

I also wonder about my second example, for CS. Does it apply--you tell me!

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

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

>

> > I just said "any shape that tiles the plane". Certainly, I've

also

> > tended to impose convexity on top of the PB property whenever a

JI,

> > untempered scale is meant.

>

> Does "shape" entail connectedness, or can it be scattered islands

> all over the place?

The latter. Especially as preimages of ETs, such constructs would be

just fine.

> I also wonder about my second example, for CS. Does it apply--you >

tell me!

I'll look at it!

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

> 1--9/8--5/4--4/3--1024/675--5/3--15/8

>

> would be an example of a scale which is CS, but neither epimorphic

> nor PB.

OK you're probably right, but epimorphic does still look like PB, and

all the examples that were ever made by Wilson, Grady, et. al., who

introduced the CS terminology, were PBs. So it is them I am thinking

about when I say CS.

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

> > Does "shape" entail connectedness, or can it be scattered islands

> > all over the place?

>

> The latter. Especially as preimages of ETs, such constructs would be

> just fine.

Under that definition, PB <==> epimorphic. Are you sure it is the accepted one?

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

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

>

> > > Does "shape" entail connectedness, or can it be scattered

islands

> > > all over the place?

> >

> > The latter. Especially as preimages of ETs, such constructs would

be

> > just fine.

>

> Under that definition, PB <==> epimorphic. Are you sure it is the >

accepted one?

The only published articles on PBs are Fokker's. Inferring strict

definitions from these articles would suggest that a parallelepiped

(or N-dimensional equivalent) are the only accepted shape (thus I

call these _Fokker_ periodicity blocks, or FPBs), and that if there

is an even number of notes, one needs to produce two alternative

versions so that symmetry about 1/1 is maintainted.

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

> The only published articles on PBs are Fokker's. Inferring strict

> definitions from these articles would suggest that a parallelepiped

> (or N-dimensional equivalent) are the only accepted shape (thus I

> call these _Fokker_ periodicity blocks, or FPBs), and that if there

> is an even number of notes, one needs to produce two alternative

> versions so that symmetry about 1/1 is maintainted.

That seems to say there *is* no accepted definition. It seems to me we want to consider at least two distinct properties--epimorphic and convex. Connectedness is another one, which would need to be defined, since more than one choice might be made there.

Hi Gene and Paul,

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, December 25, 2001 11:35 PM

> Subject: [tuning-math] Re: The epimorphic property

>

>

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

>

> > The only published articles on PBs are Fokker's. Inferring strict

> > definitions from these articles would suggest that a parallelepiped

> > (or N-dimensional equivalent) are the only accepted shape (thus I

> > call these _Fokker_ periodicity blocks, or FPBs), and that if there

> > is an even number of notes, one needs to produce two alternative

> > versions so that symmetry about 1/1 is maintainted.

>

> That seems to say there *is* no accepted definition.

If there isn't, then we're on our way to getting one. I've already

added this paragraph of Paul's to my Tuning Dictionary entry for

"periodicity-block" <http://www.ixpres.com/interval/dict/pblock.htm>,

as well as a new entry for "Fokker Periodicity Block"

<http://www.ixpres.com/interval/dict/fpb.htm>.

> It seems to me we want to consider at least two distinct

> properties--epimorphic and convex. Connectedness is another one,

> which would need to be defined, since more than one choice might

> be made there.

I'll probably add this to the definition too, unless you guide

me into what should and shouldn't be included in existing Dictionary

entries pertaining to the subject, or into what new definitions

need to be added.

-monz

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

> I'll probably add this to the definition too, unless you guide

> me into what should and shouldn't be included in existing Dictionary

> entries pertaining to the subject, or into what new definitions

> need to be added.

What I would suggest is that rather than adding something immediately, we agree on some precise definitions. To start with, does Paul see convexity or connectedness as important, as I do?

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, December 26, 2001 12:29 AM

> Subject: [tuning-math] Re: The epimorphic property

>

>

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

>

> > I'll probably add this to the definition too, unless you guide

> > me into what should and shouldn't be included in existing Dictionary

> > entries pertaining to the subject, or into what new definitions

> > need to be added.

>

> What I would suggest is that rather than adding something immediately,

> we agree on some precise definitions. To start with, does Paul see

> convexity or connectedness as important, as I do?

Thanks, Gene. In addition, I'd like to start adding some definitions

of Pierre Lamothe's terms, if he or any of you could help with those.

-monz

>

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>

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

> That seems to say there *is* no accepted definition. It seems to me

>we want to consider at least two distinct properties--epimorphic and

>convex.

Yup.

>Connectedness is another one, which would need to be defined, since

>more than one choice might be made there.

Yup -- 7-limit connected and 9-limit connected are two possibilities

in the 3D case. Some of Fokker's blocks were disconnected despite

being convex.

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

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

>

> > I'll probably add this to the definition too, unless you guide

> > me into what should and shouldn't be included in existing

Dictionary

> > entries pertaining to the subject, or into what new definitions

> > need to be added.

>

> What I would suggest is that rather than adding something

>immediately, we agree on some precise definitions. To start with,

>does Paul see convexity or connectedness as important, as I do?

I see them as important if the block is left in JI. But again, even

some of Fokker's blocks were disconnected (though still convex).

There's nothing wrong with saying "convex periodicity block" or "9-

limit-connected periodicity block" or whatever when that's the

concept you want.

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

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

>

> > The only published articles on PBs are Fokker's. Inferring strict

> > definitions from these articles would suggest that a parallelepiped

> > (or N-dimensional equivalent) are the only accepted shape (thus I

> > call these _Fokker_ periodicity blocks, or FPBs), and that if there

> > is an even number of notes, one needs to produce two alternative

> > versions so that symmetry about 1/1 is maintainted.

>

> That seems to say there *is* no accepted definition. It seems to me we want to consider at least two distinct properties--epimorphic and convex. Connectedness is another one, which would need to be defined, since more than one choice might be made there.

Gene,

Am trying to figure out this "connectedness" business

which you have been referring to in recent posts.

Followed the threads, but cannot find specifics.

Any chance that you could explain how you have

(in posting the various scales) characterized

(and subsequently quantized) what you refer to

as "connectedness"?

Curiously, J Gill

--- In tuning-math@y..., "unidala" <JGill99@i...> wrote:

> Any chance that you could explain how you have

> (in posting the various scales) characterized

> (and subsequently quantized) what you refer to

> as "connectedness"?

I took the standard set of consonant intervals in the 5, 7 and 11 limits, and defined a graph (in the sense of a simple graph from the mathematical theory of graphs) on octave equivalence classes by taking the verticies of the graph to be the tones, and the edges of the graph (which you can think of as a line drawn between two verticies) as the consonant intervals connecting the tones. "Connectivity" is simply the standard edge-connectivity of graph theory, meaning the number of edges which would need to be removed from the graph in order to render it disconnected.