Yahoo Tuning Groups Ultimate Backup tuning-math For Joe--proposed definitions

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 nth 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 p=
^ep 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.

🔗monz <joemonz@yahoo.com>

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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🔗genewardsmith <genewardsmith@juno.com>

🔗monz <joemonz@yahoo.com>

🔗manuel.op.de.coul@eon-benelux.com

🔗paulerlich <paul@stretch-music.com>

🔗genewardsmith <genewardsmith@juno.com>

--- 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 nth 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.

🔗genewardsmith <genewardsmith@juno.com>

🔗paulerlich <paul@stretch-music.com>

🔗Pierre Lamothe <plamothe@aei.ca>

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

🔗monz <joemonz@yahoo.com>

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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🔗Pierre Lamothe <plamothe@aei.ca>

🔗monz <joemonz@yahoo.com>

🔗genewardsmith <genewardsmith@juno.com>

--- 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 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.

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?

> 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

🔗Pierre Lamothe <plamothe@aei.ca>

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