Yahoo Tuning Groups Ultimate Backup tuning Interval vs interval (was mathematical course)

Paul Erlich wrote,

<< some (like Pierre Lamothe) feel that pitch heights
should correspond to simple ratios; others (like
myself) subscribe instead to the view that intervals
should correspond as closely as possible to simple
ratios; >>

I wrote,

<< I would like to temperate this impression resulting
probably of my intensive simple ratios using.

[...]

I "feel" that precision in tuning is much more important
for those who have the corresponding sensibility and pay
attention to the tuning values.

[...]

I "feel" that there exist two simultaneous perception
type of tones relation : width relations and sonance
relations. I "feel" that simple octave ratios are important
in width-axis organization while simple frequency ratios are
important in sonance-axis organization. The problem is :
these two types of simple ratios are not strictly compatible.
I never talk against temperament using and tended, in french
text, to justify its usage. [...] >>

Paul Erlich wrote,

<< Even in discussing pure JI scales, it appears we have
a significant philosophical difference. Your guiding
principle seems to be that pitch heights should correspond
to simple ratios. My guiding principle is instead that
intervals should correspond to simple ratios. >>

I wrote,

<< In all I have written in french or in english, in the
List or in private, I never talk about pitches but about
intervals.

A tone for me is a dyad interval and what I study is the
internal composition law on intervals set (axiomatic
structures). Pitches are not composable in mathematical
sense. The study of permutations on N objects has nothing
to do with the objects but with the operations on the
objects. Intervals are composable for they are like
operations. >>

Paul Erlich wrote,

<< Well that is where we differ. A tone for me is a monad (?).
A dyad interval is composed of two tones. It makes no sense
to ask what the sonance value is for the third note of the
major scale -- an isolated note doesn't have a sonance value.
One needs two notes to evaluate dyadic sonance. >>

It becomes almost ridiculous! It's like you would have pretended we differ
for you think the moon is a sphere while I think it is a cube. If you
cumulate Lapalice's thruths as you would cumulated proofs that the moon is
a sphere, would have I to conclude that I think the moon is a cube?

How can you pretend you know better than me what I think?

If you think really that I have a contradiction in my ideas, then read what
I wrote and take time to show that?

Why do you jump on a possible ambiguous formulation whose your
interpretation is clearly contradictory with ideas expressed?

Are we exchanging ideas or words?

We could say about rational numbers that they are monads (in your sense)
and also that they are precisely defined as integer dyad classes. Is it a
contradiction? I could also demonstrate that, contrary to the ambiant
numerical fetichism, a number in itself is almost nothing and has sense
only as element in a set having an operative structure, so the term monad
would be philosophicaly very inappropriate. It would be yet contradiction?

If you need to be in opposition I can suggest many topics for that. But
please try to contradict what I say and not what you wish I would have said.

Since 40 years I have never been ungulfed in the acoustical paradigm and I
worked only within the relational paradigm. Pitches and tuning problems are
outside of my views. I don't talk about notes but about intervals. I don't
talk about scale, which is composed of elements having relations, but about
a relational structure, which is a whole having modes.

I pretend I know really what means think-freely-in-interval-terms without
tuning preoccupations. I'm very glad for you if you think really in
interval terms. In that case I would invite you to search the distinction
between microtonality and macrotonality.

-----

The first step could be to distinguish at microtonal level the

P I T C H H E I G H T S

you pretend I talked about and the

I N T E R V A L W I D T H

I always use to designate one of the two dimensions
of the interval, the other being the

I N T E R V A L S O N A N C E

-----

Sorry having to write that kind of post.

Pierre Lamothe

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Pierre wrote,

>It becomes almost ridiculous! It's like you would have pretended we differ
>for you think the moon is a sphere while I think it is a cube. If you
>cumulate Lapalice's thruths as you would cumulated proofs that the moon is
>a sphere, would have I to conclude that I think the moon is a cube?

Uhh . . .

>How can you pretend you know better than me what I think?

I would make no such pretense.

> If you think really that I have a contradiction in my ideas,

I never said you have a contradiction in your ideas -- only that your ideas
and my ideas contradict one another.

> then read what
> I wrote and take time to show that?

OK -- here is an example:

On your page http://www.aei.ca/~plamothe/asymetrie.htm
<http://www.aei.ca/~plamothe/asymetrie.htm> , you write (Google
translation):

"This constitutes a weighty argument in favour of the recognition of the
sonance like consistent property of the intervals as well as that of width.
Moreover the sonance and the width are defined in a reciprocal way. For an
irreducible ratio a/b the sonance corresponds to log (ab and the width with
log (a/b Then, if one draws up the card of the first tone rational by laying
out the width abcisse some and the sonance in ordinate, one can observe
interesting musical properties. I noted, in the figure opposite, the tone of
Zarlino and the range blues. In addition, it is found there that two
families of parallel straight lines which connect much tone. One corresponds
to the major harmony (of the continuations of harmonics) and the other to a
dual harmony (of the continuations of subharmonics).

But most significant is this one: on the basis of the tone more consonant,
and while connecting gradually, with more consonant on the left then with
more consonant on the right, one constitutes a structure called tree of
Stern-Brocot (in fact it is here only about the portion of the first
octave). One connait several related structures with this one, of which
circles of Ford, and especially a monoid of modular matrices whose addition
of the columns generates the tree of Stern-Brocot, and the addition of the
lines, the tree of Euclide This last concretizes celebrates it algorithm of
the same name. "

Are you not, here, considering _tones_ to be more or less consonant, and
considering scales in terms of their _pitch-height ratios_ a/b, and
considering log(a*b) as a measure of the sonance of these tones? The graph
also makes it appear as if that is what you're doing.

>Why do you jump on a possible ambiguous formulation whose your
>interpretation is clearly contradictory with ideas expressed?

I don't understand that sentence.

>We could say about rational numbers that they are monads (in your sense)
>and also that they are precisely defined as integer dyad classes. Is it a
>contradiction?

No.

>I could also demonstrate that, contrary to the ambiant
>numerical fetichism,

Ooh . . . kinky!

>a number in itself is almost nothing and has sense
>only as element in a set having an operative structure, so the term monad
>would be philosophicaly very inappropriate. It would be yet contradiction?

I don't know what you're asking.

>If you need to be in opposition I can suggest many topics for that. But
>please try to contradict what I say and not what you wish I would have said

Pierre, my friend, I would never want to do that! I just see a difference in
philosophy between you and me. This isn't political philosophy, so I see no
need for any harsh feelings to result from these differences. This should be
like a friendly game of chess!

>I don't talk about notes but about intervals.

Perhaps something is getting lost in the translation?

>The first step could be to distinguish at microtonal level the

> P I T C H H E I G H T S

>you pretend I talked about and the

> I N T E R V A L W I D T H

>I always use to designate one of the two dimensions
>of the interval, the other being the

> I N T E R V A L S O N A N C E

As best I can understand from your web pages and many posts, your use of the
term "width" is often synonymous with my use of the term "pitch height". I'm
not pretending anything. Misunderstanding, perhaps, but not pretending.

Highest regards,
Paul

🔗Pierre Lamothe <plamothe@aei.ca>

Paul,

You wrote,

<< Pierre, my friend, I would never want to do that! I just see a
difference in
philosophy between you and me. This isn't political philosophy, so I see no
need for any harsh feelings to result from these differences. This should be
like a friendly game of chess! >>

I apologize for the harsh tone consecutive to a frustration but be sure my
deep feelings don't vary as my tone. Besides I'm not frustrated by our
differences (which lead to enrichment) but by the fact I don't recognize my
ideas in some descriptions of what I should think.

However I accept I have the responsability to find the adequate language to
be understood. I will take time to search the source leading to some
confusions and I will continue to simplify my approach in order to be
understood by more readers.

You wrote,

<< As best I can understand from your web pages and many posts, your use of
the
term "width" is often synonymous with my use of the term "pitch height". >>

I am very astonished that I could have given the impression that "width"
would be synonymous with "pitch height". Since I have a certain knowledge
in physics, you understand certainly I could hardly confuse a
non-dimensional "width" and a dimensional "pitch height" measured in Hertz.

----------

Now, I would like to verify if the translation of "ton" by "tone" could be
a source of confusion :

I presume the term "tone" in English corresponds to an element in a scale
whose ratio is f/t where f is the frequency of the corresponding note and t
the frequency of the tonic. In that sense "tone" would refer to a unique
interval (between the tonic and the note) and not to a class of intervals.

In French, I had defined "ton" (what is translated by tone) as an interval
of the first octave representing the class of intervals modulo 2.

More precisely :

Let G = <2 p q .. > ZxZxZ .. be the group or rational intervals
corresponding to the primal basis <2 p q .. > and G/2 be the quotient-group
of G by the subgroup having the primal basis <2> Z. The elements of G/2 are
the classes of intervals modulo 2.

The subgroup <p q .. > ZxZ .. and the subgroup corresponding to the
intersection of G with [1,2[ (the first octave) are two good representing
systems of G/2 since they have exactly one element in each class.

By definition, I name in French

" pivot " an element of the first subgroup, and

" ton " an element of the second subgroup

both as representing element of their own class modulo 2.

e.g. In a group containing the subgroup <2 3 5> ZxZxZ

40 and 5/128 are in the same class modulo 2
represented by the "pivot" 5 and the "ton" 5/4

48/5 and 3/80 are in the same class modulo 2
represented by the "pivot" 3/5 and the "ton" 6/5

I also used sometimes "ton" in the sense of step with the same ambiguity we
find in terms like diatonic (steps) and heptatonic (degrees).

Maybe I should never use the term tone in English to designate my concept
of "ton".

Highest regards,

Pierre

🔗jpehrson@rcn.com

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

/tuning/topicId_20890.html#20940

>
> Now, I would like to verify if the translation of "ton" by "tone"
could be a source of confusion :
>
> I presume the term "tone" in English corresponds to an element in a
scale whose ratio is f/t where f is the frequency of the
corresponding note and t the frequency of the tonic. In that sense
"tone" would refer to a unique interval (between the tonic and the
note) and notto a class of intervals.

Hello Pierre...

I don't believe "tone" in English could ever refer to anything other
than a single note. It could NEVER be an interval, of whatever
nature...

>
> In French, I had defined "ton" (what is translated by tone) as an
interval of the first octave representing the class of intervals
modulo 2.
>
> More precisely :
>
> Let G = <2 p q .. > ZxZxZ .. be the group or rational intervals
> corresponding to the primal basis <2 p q .. > and G/2 be the
quotient-group of G by the subgroup having the primal basis <2> Z.
The elements of G/2 are the classes of intervals modulo 2.
>
> The subgroup <p q .. > ZxZ .. and the subgroup corresponding to the
> intersection of G with [1,2[ (the first octave) are two good
representing systems of G/2 since they have exactly one element in
each class.
>
> By definition, I name in French
>
> " pivot " an element of the first subgroup, and
>
> " ton " an element of the second subgroup
>
> both as representing element of their own class modulo 2.
>
> e.g. In a group containing the subgroup <2 3 5> ZxZxZ
>
> 40 and 5/128 are in the same class modulo 2
> represented by the "pivot" 5 and the "ton" 5/4
>
> 48/5 and 3/80 are in the same class modulo 2
> represented by the "pivot" 3/5 and the "ton" 6/5
>
> I also used sometimes "ton" in the sense of step with the same
ambiguity we find in terms like diatonic (steps) and heptatonic
(degrees).
>

> Maybe I should never use the term tone in English to designate my
concept of "ton".
>

I believe that is correct. The two terms are definitely NOT
equivalent....

Thanks!

________ _______ ______ ____
Joseph Pehrson

🔗Pierre Lamothe <plamothe@aei.ca>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Hi Pierre,

>I am very astonished that I could have given the impression that "width"
>would be synonymous with "pitch height". Since I have a certain knowledge
>in physics, you understand certainly I could hardly confuse a
>non-dimensional "width" and a dimensional "pitch height" measured in Hertz.

Pitch height doesn't have to be measured in Hertz. It can be measured in
cents, or as a ratio relative to an arbitrary tonic. It is in the latter
sense that I understand your "widths".

>I presume the term "tone" in English corresponds to an element in a scale
>whose ratio is f/t where f is the frequency of the corresponding note and t
>the frequency of the tonic. In that sense "tone" would refer to a unique
>interval (between the tonic and the note) and not to a class of intervals.

It could refer to a class of intervals between the tonic and the note.

>In French, I had defined "ton" (what is translated by tone) as an interval
>of the first octave representing the class of intervals modulo 2.

I understood the rest of your message. The "class modulo 2" part should not
be a problem in the translation between French and English. The problem is
"interval". "Interval" requires two notes. One of them can be the tonic, if
there even is a tonic. But maybe neither is the tonic. If you're always
measuring the interval upwards from the tonic, then all you're doing is
characterizing the pitch-height (or pitch-class, which is pitch-height
modulo 2) of a tone.

In my opinion, the only conditions under which your "intervals", which I
call "pitch heights", correspond to a sonance measure in the way you
illustrate, is if the music is in two voices, and one voice holds the tonic
_always_. Then each tone-ratio will also result in the identical
interval-ratio, and sonance measure will be correct. However, precious
little music is in this form.

And in the pre-tonal era, it can sometimes be next to impossible to assign
one of the pitches as the "tonic". Yet it is undeniable that sonance of
intervals was a powerful feature of the diatonic scale in Medieval and
Renaissance music.

I stop now in order to reiterate that this is just my view and to allow for
responses.

Oops -- one more thing.

>I also used sometimes "ton" in the sense of step with the same ambiguity we
>find in terms like diatonic (steps) and heptatonic (degrees).

I think you might be misunderstanding the etymology of the term diatonic.
The term comes from the prefix dia-, meaning "through", and _not_ from the
prefix di-, meaning two. "Diatonic" simply means "through the tones" --
specifically, through the tones of the ordinary heptatonic scale. The only
difference between "diatonic" and "heptatonic" is that "diatonic" is a bit
more specific, while any 7-tone scale can be "heptatonic". At least in
English that's what it means.

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Joseph Pehrson wrote,

>Hello Pierre...

>I don't believe "tone" in English could ever refer to anything other
>than a single note. It could NEVER be an interval, of whatever
nature...

With all due respect, Joseph, I must disagree. Pierre specifically said that
it refers to the interval between the tonic and the note. What on earth is
wrong with saying that the tones of the Zarlino JI scale are

C D E F G A B C'
1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1?

These are the intervals between the tonic and the tones, yes, but they can
also be understood to simply name the tones themselves. In fact, on this
list we've tried, many times, to adopt the convention that a ratio written
with a slash ("a/b") refers to a _tone_ or a _pitch_, with the ratio giving
the relationship between that tone and the tonic; while a ratios written
with a colon ("a:b") refers to an interval between two tones. For example,
the complete list of intervals in the Zarlino JI scale is

------------upper note---------------
C D E F G A B C'

1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1 C }
1:1 10:9 32:27 4:3 40:27 5:3 16:9 D }
1:1 16:15 6:5 4:3 3:2 8:5 E }
1:1 9:8 5:4 45:32 2:1 F } lower
1:1 10:9 5:4 4:3 G } note
1:1 9:8 6:5 A }
1:1 16:15 B }
1:1 C'}

Pierre, if you're reading this, it is my philosophy that sonance (measured,
for example, by a*b) is an attribute that only applies when the ratio of a
to b refers to an interval ("a:b") and never to a tone ("a/b").

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Hi Pierre,

As I just explained, I completely disagree with Joseph's reply to you:

><< I don't believe "tone" in English could ever refer to
> anything other than a single note. It could NEVER be
> an interval, of whatever nature... >>

Now you write,

>I should have had to verify that before and it might be a good thing I ask
>advice in future about words I could suspected.

In this case, I believe your initial instinct was correct.

>In such case, I would ask to you and other members your advice about the
>expression

> "interval modulo 2" (what almost defines my "ton" in French).

I don't think that is what you want to use. If the Zarlino JI heptatonic
scale is extended over several octaves, then its intervals modulo 2 would
be:

------------upper note-----------
C D E F G A B

1:1 9:8 5:4 4:3 3:2 5:3 15:8 C }
16:9 1:1 10:9 32:27 4:3 40:27 5:3 D }
8:5 9:5 1:1 16:15 6:5 4:3 3:2 E }
3:2 27:16 15:8 1:1 9:8 5:4 45:32 F } lower
4:3 3:2 5:3 16:9 1:1 10:9 5:4 G } note
6:5 27:20 3:2 64:45 9:5 1:1 9:8 A }
16:15 6:5 4:3 64:45 8:5 16:9 1:1 B }

On the other hand, the pitches modulo 2 of the Zarlino JI heptatonic scale
are

C D E F G A B
1/1 9/8 5/4 4/3 3/2 5/3 15/8

And it is the latter that you mean by "ton", correct?

Respectfully,
Paul

🔗monz <MONZ@JUNO.COM>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20890.html#20956

> Hello Pierre...
>
> I don't believe "tone" in English could ever refer to anything
> other than a single note. It could NEVER be an interval, of
> whatever nature...

Joe, be careful not to make broad assumptions like this.

In many medieval treatises, the Latin word "tonus", which
would normally be translated simply as "tone" in English,
is used to describe not only a single pitch, nor only an
interval, but an entire modal scale.

And "tone" *has* been used in this sense later, in English,
during the 1800s in various American JI solfege schemes.

"Psalm tone" is a similar ecclesiastical usage, refering to
an entire melodic chant sequence.

I realize that all three of these examples would be encountered
far less frequently than the usual definition you illustrate.
But they do exist and are occasionally found.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_20890.html#20975

> Joseph Pehrson wrote,
>
> >Hello Pierre...
>
> >I don't believe "tone" in English could ever refer to anything
other than a single note. It could NEVER be an interval, of
whatever
> nature...
>

> With all due respect, Joseph,

Actually, less is probably due than you might suspect...

>I must disagree. Pierre specifically said that it refers to the
interval between the tonic and the note. What on earth is
> wrong with saying that the tones of the Zarlino JI scale are
>
> C D E F G A B C'
> 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1?
>
> These are the intervals between the tonic and the tones, yes, but
they can also be understood to simply name the tones themselves. In
fact, on this list we've tried, many times, to adopt the convention
that a ratio written with a slash ("a/b") refers to a _tone_ or a
_pitch_, with the ratio giving the relationship between that tone and
the tonic; while a ratios written with a colon ("a:b") refers to an
interval between two tones.

Well, it does seem like a pretty good convention... David Doty, of
course, advocates it as well in the JI Primer...

However, the RESULT is still ONE sounding "thingy" a pitch, or TONE,
right... even if it's "theoretical" basis refers to its relationship
to the tonic... Or am I misunderstanding something.

In any case, the notion of "tone" is different from the broad
categories of "sonance" that Pierre is discussing in his posts...

Or, perhaps I was just reading the French wrong?? Quite possibly.

_______ _____ _____ ___
Joseph Pehrson

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Joseph wrote,

>However, the RESULT is still ONE sounding "thingy" a pitch, or TONE,
>right... even if it's "theoretical" basis refers to its relationship
>to the tonic...

EXACTLY . . . this is why I disagreed with you when you suggested that
Pierre should translate his use of the word "ton" to "interval".

>In any case, the notion of "tone" is different from the broad
>categories of "sonance" that Pierre is discussing in his posts...

Well I think that's a matter of the philosophical difference I've been
speaking of . . . I think Pierre sees all this in a much more abstract way
than you and I . . . but let me not presume to speak for Pierre . . . see
what you can get from his reply to me . . .

>Or, perhaps I was just reading the French wrong?? Quite possibly.

I wish I had the time to work on my French and try to understand Pierre in
his own language . . . I'm sure it would help matters greatly.