Scale

A discrete set of real numbers, containing 1, and such that the distance be=

tween sucessive elements of the scale is bounded both below and above by po=

sitive real numbers. The least upper bound of the intervals between successi=

ve elements of the scale is the maximum scale step, and the greatest lower b=

ound is the minimum scale step. The element of the scale obtained by countin=

g up n scale steps is the nth degree, by counting down is the Βnth degree; 1=

is the 0th degree.

Tone group

A set of positive real numbers closed under multiplication and inversion (s=

o that if x is in the set, so is 1/x), and regarded as a set of intervals or=

pitches.

Rational tone group

A tone group whose elements are rational numbers.

Val

A map from a rational tone group to the integers, which respects multiplica=

tion. If h is a val, then h(a*b) = h(a) + h(b); h(1) = 0; and h(1/a) = -h(a)=

. If we write the rational number "a"as

a = 2^e2 * 3^e3 * ... * p^ep, we may denote it by a row vector

[e2, e3, ...., ep]. In that case we denote a val by a column vector of inte=

gers of the same dimension. In the language of abstract algebra, a val is a =

homomorphism from the tone group to the integers.

Canonical val

For any positive integer n, the canonical val hn is the val such that

hn(p) = round(n * log2(p)), where p is prime and where "round" means round =

to the nearest integer. The restriction of hn to a particular rational tone =

group is also denoted hn.

P-adic val

For any prime p, the p-adic val vp is the val which sends q = 2^e2 3^e3 Β
p=

^ep Β
to vp(q) = ep. This is called the "p-adic valuation" in number theory.=

Epimorphic

A scale has the epimorphic property, or is epimorphic, if there is a val h =

such that if qn is the nth scale degree, then h(qn)=n. The val h is the char=

acterizing val of the scale.

Thanks, Gene! This is great. One quibble...

> From: genewardsmith <genewardsmith@juno.com>

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

> Sent: Thursday, January 10, 2002 8:11 PM

> Subject: [tuning-math] For Joe--proposed definitions

>

>

> Scale

>

> A discrete set of real numbers, containing 1, and such

> that the distance between sucessive elements of the scale

> is bounded both below and above by positive real numbers.

> The least upper bound of the intervals between successive

> elements of the scale is the maximum scale step, and the

> greatest lower bound is the minimum scale step. The element

> of the scale obtained by counting up n scale steps is the

> nth degree, by counting down is the -nth degree; 1 is the

> 0th degree.

These definitions are necessary for helping a reader to

understand your tuning-math posts.

While the definitions for "Tone group" and "Val" and their

derivitives, and "Epimorphic", seem adequate to me, the one

for "Scale", however, requires more than just your mathematical

lingo.

The fact that the Tuning Dictionary has been online for 3 years

without a definition of "scale" attests to my procrastination in

coming up with a good definition of it. It's been a real gap

in the Dictionary, possibly the most important tuning term of

all.

This definition is great for understanding your work, but

is it really correct to simply *define* a scale as "a discrete

set of real numbers"? I'm thinking that this is a Dictionary

entry which, like many of them, will have numbered definitions

to denote distinctions in meaning. The purely musical definition

of "scale" would have to begin with "... a set of musical

*pitches* ...", etc. Yours would be a #2 definition.

Paul, since you're a mathematically-aware musician,

can you help with this?

-monz

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

The purely musical definition

> of "scale" would have to begin with "... a set of musical

> *pitches* ...", etc. Yours would be a #2 definition.

I was aware it wouldn't fit all usages. I wanted a defintion which was precise enough to use mathematically, and this is a start. In fact, you might want to be more restrictive and not allow a lot of the things I am calling scales here--that could be discussed.

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

> Thanks, Gene! This is great. One quibble...

>

> > Scale

> >

> > A discrete set of real numbers, containing 1, and such

> > that the distance between sucessive elements of the scale

> > is bounded both below and above by positive real numbers.

Incidentally, this is incorrect as written since we need to define "distance" logarithmically. If we take a scale as defined above and send scale element s to 2^s, then we get a scale considered as pitch values. In other words, these scale elements should be considered as cents or something of that sort. I'll rewrite it if you like, but perhaps we should hear from other people first.

> From: genewardsmith <genewardsmith@juno.com>

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

> Sent: Thursday, January 10, 2002 11:21 PM

> Subject: [tuning-math] Re: For Joe--proposed definitions

>

>

> [re: definition of "scale"]

> I'll rewrite it if you like, but perhaps we should hear

> from other people first.

Can I have both? :) I'd like to have the input of

others (especially Paul, Dave, Graham, Manuel) on this.

But feel free to rewrite it as you see fit.

I'm not yet up to speed with your work here over the past

few months, so I really can't comment until I understand more.

-monz

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I've never felt the need for a mathematical definition

of "scale". Never looked it up in a dictionary either.

Perhaps "things like do re mi fa sol la ti do" will do.

Manuel

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

> Thanks, Gene! This is great. One quibble...

>

>

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

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

> > Sent: Thursday, January 10, 2002 8:11 PM

> > Subject: [tuning-math] For Joe--proposed definitions

> >

> >

> > Scale

> >

> > A discrete set of real numbers, containing 1, and such

> > that the distance between sucessive elements of the scale

> > is bounded both below and above by positive real numbers.

> > The least upper bound of the intervals between successive

> > elements of the scale is the maximum scale step, and the

> > greatest lower bound is the minimum scale step. The element

> > of the scale obtained by counting up n scale steps is the

> > nth degree, by counting down is the -nth degree; 1 is the

> > 0th degree.

>

>

> These definitions are necessary for helping a reader to

> understand your tuning-math posts.

Are they? I would think that they would be useful mainly for a

mathematician who may not know anything about music but who still may

wish to understand Gene's research.

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

> Can I have both?

Scale

A discrete set of real numbers, containing 0, and regarded as defining tone=

s in a logarithmic measure, such as cents or octaves,

and such that the distance between sucessive elements of the scale is bound=

ed both below and above by positive real numbers. The least upper bound of t=

he intervals between successive elements of the scale is the maximum scale s=

tep, and the greatest lower bound is the minimum scale step. The element of =

the scale obtained by counting up n scale steps is the nth degree, by counti=

ng down is the Βnth degree; 0 is the 0th degree. The set of positive real nu=

mbers which are the tones so represented is also regarded as the scale.

Periodic scale

If s(n) denotes the nth degree of a scale, it is *periodic* with period L>0=

if there is an integer k>0 such that s(n+k) = s(n) + L.

The most common value of L is the octave, which defines octave periodic sca=

les.

Period reduced scale

A period reduced scale with period L is the unique set of representatives i=

n the range 0<=s<L of a periodic scale with period L.

Connected scale

A scale is connected with respect to a set of intevals S if for any

two scale degrees a and b, there is a path a=a_0, a_1, ... a_n = b

such that |a_i - a_{i+1}| is an element of S.

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

> Are they? I would think that they would be useful mainly for a

> mathematician who may not know anything about music but who still may

> wish to understand Gene's research.

They are useful for anyone (eg, me) who might want to state and prove theorems. You can't do that without definitions.

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

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

>

> > Are they? I would think that they would be useful mainly for a

> > mathematician who may not know anything about music but who still

may

> > wish to understand Gene's research.

>

> They are useful for anyone (eg, me) who might want to state and

>prove theorems. You can't do that without definitions.

Exactly. Note I said _may_.

I think it would be good not only to separate math definitions but also to avoid using

superficial blending of tuning notions with math concepts in terms like Tone group

and Rational tone group.

Gene wrote:

Tone group

A set of positive real numbers closed under multiplication and inversion

(so that if x is in the set, so is 1/x), and regarded as a set of intervals or

pitches.

How could you regard an infinite group as a set of intervals or pitches? If a such regard

implies the restriction to the audible pitches, or the intervals between distinct audible

pitches, this mathematical object loose its group structure, unless to be a tempered

monogenic group.

It is important to understand, from the modelizing viewpoint, that any group generated

by more than one rational number, say 2 and 3, would have an infinite number of tones

in the octave. The group structure may not modelize the operative structure on a finite

set of rational intervals.

I consider this group as a simple mathematical object, useful as tool but inadequate as

model. That don't prevent to plunge a finite set of tones in this group when needed but

to extend the mathematical properties of the group structure as it would represent the

proper operative structure on the given tones.

In my opinion, any attempt to understand a categorical perception of the tones in a

musical context without reference to an underlying proper invariant operative structure,

confined to illusion. What distinguishes a perception from a sensation, or an impression,

is that the conscious part is significative (has a musical sense for the auditor) while the

automated ability part to recognize a significative intention was learned mainly from the

habitual physionomy of the transformations. An isolated experience focusing on sensation

is a regressive perception and attempts to reconstruct an authentic musical experience

with such isolated acoustic impressions appears to me well superficial.

Gene wrote:

Canonical val

For any positive integer n, the canonical val hn is the val such that

hn(p) = round(n * log2(p)), where p is prime and where "round" means

round to the nearest integer. The restriction of hn to a particular rational

tone group is also denoted hn.

It is important to understand that Canonical val don't mean unique pertinent val in

musical sense. For instance, the canonical val

h5(3) = 8

h5(5) = 12

corresponds to the chinese system (gammier 1)

1 4 2

4 2 0 3 1

. . 3 1 4

while the equally valid japanese system (gammier 8)

. 3 1 4

4 2 0 3 1

. 1 4 2

corresponds to the non canonical variant

h5(5) = 11

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

Some remarks now about the proposed definition for the tuning notion of scale.

Gene wrote first:

Scale

A discrete set of real numbers, containing 1, and such that the distance

between sucessive elements of the scale is bounded both below and above

by positive real numbers. The least upper bound of the intervals between

successive elements of the scale is the maximum scale step, and the

greatest lower bound is the minimum scale step. The element of the scale

obtained by counting up n scale steps is the nth degree, by counting down

is the –nth degree; 1 is the 0th degree.

and corrected with:

Scale

A discrete set of real numbers, containing 0, and regarded as defining tones

in a logarithmic measure, such as cents or octaves, and such that the distance

between sucessive elements of the scale is bounded both below and above

by positive real numbers. The least upper bound of the intervals between

successive elements of the scale is the maximum scale step, and the greatest

lower bound is the minimum scale step. The element of the scale obtained by

counting up n scale steps is the nth degree, by counting down is the –nth

degree; 0 is the 0th degree. The set of positive real numbers which are the tones

so represented is also regarded as the scale.

Gene, is there the norm of elegance you had in head when you criticized my work at

Christmas day?

I can understand why Manuel said there is no need for a mathematical scale definition.

However I would say that at first level, in the reduced minimal sense he uses, the scale

has essentially a mathematical definition as

discrete subset on an ordered set

Besides, I can understand the procrastination of Monz to attack this basic definition.

Intuitive as he is, I would believe that he feels its inherent difficulties.

The next level implies the interval notion which requires much more than an ordered set

as starting structure. If

a - b - c - d - ...

is a such discrete ordered set, the successive steps, we would denote

a:b - b:c - c:d - ...

have sense only if there exist a composition law on that set permitting to define also an

order on the intervals. So with the < relation (resp. > or =), depending of the law type

(multiplicative or additive), we have

a:b < c:d <=> bc < ad

a:b < c:d <=> b+c < a+d

It's there a first element that is missing. I don't have intention to explain all what is implied

in a such definition, so Gene would have only to add his name. I leave him to formulate

adequate proposition, reserving my comments.

I would signal only an error for the moment: changing the neutral element from 1 to 0

means implicitely using now the additive composition law, so the term positive has no more

sense.

Pierre

Hello Pierre, and thanks for the detailed comments on

Gene's definitions. I will leave it to the more mathematically

knowledgeable to argue on the differences ... I don't really

understand them yet. But I have a few comments below ...

> From: Pierre Lamothe <plamothe@aei.ca>

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

> Sent: Saturday, January 12, 2002 3:01 AM

> Subject: [tuning-math] Re: For Joe--proposed definitions

>

>

> Besides, I can understand the procrastination of Monz

> to attack this basic definition. Intuitive as he is,

> I would believe that he feels its inherent difficulties.

I wouldn't characterize what I said as an "attack", just

an alternative perspective. My main problem is not really

with Gene's definition -- it's with the fact that *I* have

not yet written a definition for this all-important tuning

word, one that would have lots of meaning for musicians.

Certainly, composers and improvisers are often guided by

mathematical properties of scales, as when they make

"punning" games with the harmonies. But my feeling is

that when musicians are concerned mainly with *playing*

music, they're guided more by the visceral sensory perception

of the sound itself than by the numerical aspects of their

pitch-ordering and other relationships.

The fact that it's hard to talk about a musical scale

without invoking the concept of a mathematical scale is

what's caused me to hesitate so long in writing a definition,

and I think *that* kind of definition really should be in

place alongside Gene's. In simple terms, the mathematical

defintion is great, but it's not enough.

> I don't have intention to explain all what is implied

> in a such definition, so Gene would have only to add his

> name. I leave him to formulate adequate proposition,

> reserving my comments.

Pierre, you are welcome to contribute anything you would like

to the Tuning Dictionary, and whatever you submit will be

credited with your name.

BTW, I've put your definitions of "chordoid" and "gammier"

into the Dictionary exactly as you posted them.

http://www.ixpres.com/interval/dict/lamothe/chordoid.htm

http://www.ixpres.com/interval/dict/lamothe/gammier.htm

(They're the same, just linked under different names.)

But I would like to give those pages the same look as all

my other webpages. Do you have any objection?

-monz

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Hello Monz,

But I would like to give those pages the same look as all

my other webpages. Do you have any objection?

For sure, I have no objection. What is important is the sense. Besides, I would prefer also you

would correct eventually my English.

Pierre

> From: monz <joemonz@yahoo.com>

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

> Sent: Saturday, January 12, 2002 3:40 AM

> Subject: Re: [tuning-math] Re: For Joe--proposed definitions

>

>

> BTW, I've put your definitions of "chordoid" and "gammier"

> into the Dictionary exactly as you posted them.

> http://www.ixpres.com/interval/dict/lamothe/chordoid.htm

> http://www.ixpres.com/interval/dict/lamothe/gammier.htm

>

> (They're the same, just linked under different names.)

I decided that rather than separate these into two separate

definitions, they should stay together as Pierre posted them.

So the index for both terms points to "gammier.htm";

"chordoid.htm" is now a dead link.

-monz

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

> I think it would be good not only to separate math definitions but also to avoid using

> superficial blending of tuning notions with math concepts in terms like Tone group

> and Rational tone group.

We already talk about the 7-limit, 11-limit and so forth, which are tone groups, whatever one calls them. Why not give them a name? I don't understand you objection.

I suppose we could call them all regular temperaments, with the proviso that rational intonation is a temperament, but I think that would be asking for trouble.

> How could you regard an infinite group as a set of intervals or pitches?

It's done all the time implicitly. We would say "four octaves of 12 equal temperament" or "five octaves of twelve equal temperament" without regarding them as different temperaments. Keeping things as groups means keeping things simple, and we want things to be simple.

> It is important to understand, from the modelizing viewpoint, that any group generated

> by more than one rational number, say 2 and 3, would have an infinite number of tones

> in the octave. The group structure may not modelize the operative structure on a finite

> set of rational intervals.

Isn't this like saying there are an infinite number of possible lengths in a meter so we should not regard length measurements as real numbers? This just leads to confusion.

> In my opinion, any attempt to understand a categorical perception of the tones in a

> musical context without reference to an underlying proper invariant operative structure,

> confined to illusion.

I don't imagine there are any underlying invariant structures, but intervals can be recognized as transposed, and approximate small integer ratios have a clear significance; modeling that in the simplest way (with morphisms that actually illuminate the situation) leads to finitely generated groups.

An isolated experience focusing on sensation

> is a regressive perception and attempts to reconstruct an authentic musical experience

> with such isolated acoustic impressions appears to me well superficial.

I think you start from the ground up, and don't float down from the sky. Simple is good.

> Gene, is there the norm of elegance you had in head when you criticized my work at

> Christmas day?

My defintion is exactly what yours aren't and I wish they were, frankly. A definition is not supposed to be elegant, it is supposed to be precise. If it isn't, it does not do the job it needs to do, which is to convey an exact sense which allows one to understand precisely what is meant, what some is and what it isn't.

> I can understand why Manuel said there is no need for a mathematical scale definition.

If we are going to define "connected scale", "convex scale", "epimorphic scale" and so forth we first define "scale".

> However I would say that at first level, in the reduced minimal sense he uses, the scale

> has essentially a mathematical definition as

> discrete subset on an ordered set

I'm not interested in defining scales on hyperreal numbers or the long line! Real numbers are clearly what we want.

> The next level implies the interval notion which requires much more than an ordered set

> as starting structure. If

> a - b - c - d - ...

> is a such discrete ordered set, the successive steps, we would denote

> a:b - b:c - c:d - ...

> have sense only if there exist a composition law on that set permitting to define also an

> order on the intervals. So with the < relation (resp. > or =), depending of the law type

> (multiplicative or additive), we have

> a:b < c:d <=> bc < ad

> a:b < c:d <=> b+c < a+d

All of which makes me think we should at least be working in a totally ordered field, but I don't understand your point. Why all the complicated bells and whistles? We surely do not want to talk about scales which are not contained in the real numbers!

Gene wrote:

We already talk about the 7-limit, 11-limit and so forth, which are tone groups,

whatever one calls them. Why not give them a name? I don't understand

your objection..

If you talked about Tone group and Rational tone group as mathematical structures in which

you plunge the few possible tuning values, I would have few to say. But if you define these

objects as tuning concepts, without to precise that the group property is not conserved in the

restriction to the true tuning values, I say you contribute to propagate a pernicious attitude

I have already named numerical fetichism.

I suppose we could call them all regular temperaments, with the proviso that

rational intonation is a temperament, but I think that would be asking for

trouble.

Sorry, I don't get it. I suppose it might have sense in temperament language.

> How could you regard an infinite group as a set of intervals or pitches?

It's done all the time implicitly. We would say "four octaves of 12 equal

temperament" or "five octaves of twelve equal temperament" without regarding

them as different temperaments. Keeping things as groups means keeping things

simple, and we want things to be simple.

There is no problem with tempered case since the restriction to the octave conserve the

group structure. The problem is essentially in the JI case where the absence of the closure

axiom implies that none finite part is a subgroup.

> It is important to understand, from the modelizing viewpoint, that any group

> generated by more than one rational number, say 2 and 3, would have an infinite

> number of tones in the octave. The group structure may not modelize the operative

> structure on a finite set of rational intervals.

Isn't this like saying there are an infinite number of possible lengths in a meter so we

should not regard length measurements as real numbers? This just leads to confusion.

We can regard frequency measurements as real numbers and the operative properties on these

measurements as those of the real field, in acoustical calculation. But there exist approximatively

3500 internal cilied cells in the cochlea and thus no possibility to perceive really more distinct

values. Could you transport the operative properties of the real field in the space of pitch height

values?

For instance, there exist always another element between any two elements in the real field.

Is it the case between perceived pitch values?

> In my opinion, any attempt to understand a categorical perception of the

> tones in a musical context, without reference to an underlying proper invariant

> operative structure, confined to illusion.

I don't imagine there are any underlying invariant structures, but intervals

can be recognized as transposed, and approximate small integer ratios have a

clear significance; modeling that in the simplest way (with morphisms that

actually illuminate the situation) leads to finitely generated groups.

I suggest for exciting your imagination you read about perception phenomenology. Your simplest

way is simply not the way we perceive. Do you know, for instance, that the attribution of a color

to an object is not built-in but has to be learned? A born blind recently operated see first colors

without to attribute them to objects. We construct our capacity to perceive with our intentional

activities. Our aquired abilities don't depend of inaccessible universal mathematical properties but

only of those in the space of our real experimentation, by the necessity where we are to integrate,

second by second, a sensorial flux renewing constantly aspects of things we intentionaly focus on.

> Gene, is there the norm of elegance you had in head when you criticized my

> work at Christmas day?

My defintion is exactly what yours aren't and I wish they were, frankly. A

definition is not supposed to be elegant, it is supposed to be precise. If it

isn't, it does not do the job it needs to do, which is to convey an exact sense

which allows one to understand precisely what is meant, what some is and what

it isn't.

I showed to you a short presentation of the chord theorem, in French. The definitions were simple,

precise, conveying an exact sense and were elegant. But it was in French, it was not what you wish,

and you had none idea about the fertility of this theorem. Is by lazziness, as you mentioned about

number like 123.45678901234, that you prefered to criticizise? If you believe really your definition is

undoubtly a good definition, I could take time to propagate doubts.

> I can understand why Manuel said there is no need for a mathematical scale

> definition.

If we are going to define "connected scale", "convex scale", "epimorphic scale"

and so forth we first define "scale".

Yes, but you have, first, to take account that there exist already an implicit definition, for instance,

in the scale list of Scala. Secondly, ... I stop here, since you believe you can start afresh,

reformulating all concepts as yours.

> However I would say that at first level, in the reduced minimal sense he

> uses, the scale has essentially a mathematical definition as discrete subset

> on an ordered set

I'm not interested in defining scales on hyperreal numbers or the long line!

Real numbers are clearly what we want.

Where do you go with your dirty word? Do you feel bad for I don't use numbers? I talked

only about two simple mathematical properties that anybody in tuning List may understand.

Real numbers have sense in the temperament treatment but have nothing to do with perceptible

pitch height

> The next level implies the interval notion which requires much more than an

> ordered set as starting structure. If

> a - b - c - d - ...

> is a such discrete ordered set, the successive steps, we would denote

> a:b - b:c - c:d - ...

> have sense only if there exist a composition law on that set permitting to

> define also an order on the intervals. So with the < relation (resp. > or =),

> depending of the law type (multiplicative or additive), we have

> a:b < c:d <=> bc < ad

> a:b < c:d <=> b+c < a+d

All of which makes me think we should at least be working in a totally ordered

field, but I don't understand your point. Why all the complicated bells and

whistles? We surely do not want to talk about scales which are not contained in

the real numbers!

I simply note that to pass of order in an ordered set to the order of what we could name

interval between the elements of the ordered set, we need a composition law on the set.

I say that we have not to use infinity to position a very little set of tones with its relations.

Pierre

I've been archiving the dialog between Gene and Pierre over

the definitions Gene proposed recently. Here it is:

http://www.ixpres.com/interval/dict/lamothe/genemath-crit.htm

This was hastily done, and has some aesthetic flaws, such

as the difference in font styles between Gene and Pierre.

But I'd like everyone to be able to easily follow the whole

discussion, and I welcome the input of other voices

... mainly because I myself understand so little of it. :(

Here are the original definitions by Gene:

http://www.ixpres.com/interval/dict/genemath.htm

-monz

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