Limit terminology

[Paul E:]
>I'd suggest perhaps distinguishing the versions in which you
>specifically target 7:9 intervals as 9-limit, rather than 7-limit,
>though the terminology is not perfectly precise given your "tuning
>file" methodology.

I know this issue has been thrashed out repeatedly on this list, but I
must admit I still don't "get it". If in what I would call "5-limit"
tuning, B is 15/16 of C, how is that not "15-limit" according to your
definition?

If a chord includes C,G, and D, of course D is going to be tuned as
9/4 of C, but to my mind that doesn't make it "9-limit".

Perhaps it would be simpler just to qualify using the term "prime limit"
as opposed to "odd limit". Though I am aware of, and largely agree
with, the argument that "odd limt" more accurately reflects the
difficulty of tuning a complex dyad to JI by ear, when it comes to the
actual process of assigning tunings to notes, designating a prime limit
makes more sense to me.

As for my "tuning file methodology", I'm still not sure if the basic
process I use is clear. Can you enlighten me?

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Paul E:]
> >I'd suggest perhaps distinguishing the versions in which you
> >specifically target 7:9 intervals as 9-limit, rather than 7-limit,
> >though the terminology is not perfectly precise given your "tuning
> >file" methodology.
>
> I know this issue has been thrashed out repeatedly on this list, but I
> must admit I still don't "get it". If in what I would call "5-limit"
> tuning, B is 15/16 of C, how is that not "15-limit" according to your
> definition?

Because it is not the dyad C-B that is "attracted" to the ratio 15:16. Instead, the 15:16
results in one or both of the following ways:

C-E is attracted to 4:5 and E-B is attracted to 2:3
C-G is attracted to 2:3 and G-B is attracted to 4:5
>
> If a chord includes C,G, and D, of course D is going to be tuned as
> 9/4 of C, but to my mind that doesn't make it "9-limit".

Exactly . . . C-G is attracted to 2:3 and G-D is attracted to 2:3, so there is no necessary
supposition that C-D as a dyad is attracted to 4:9 (though it may be in some
circumstances -- that would be a "9-limit" situation).

So perhaps you do "get it" after all?
>
> Perhaps it would be simpler just to qualify using the term "prime limit"
> as opposed to "odd limit". Though I am aware of, and largely agree
> with, the argument that "odd limt" more accurately reflects the
> difficulty of tuning a complex dyad to JI by ear, when it comes to the
> actual process of assigning tunings to notes, designating a prime limit
> makes more sense to me.

Well, if you look at my definition of "limit" in the Monzo Dictionary, I specifically state that
the prime definition of limit is the one to use when describing JI tuning systems, and the
odd definition is the one to use when describing consonance of isolated sonorities. Since
the context of adaptively tuning pieces of music is perhaps somewhere between these
two contexts, it's no wonder the terminology gets confusing . . .

As for "11/13-limit", I still think

(a) you should just call it 13-limit.
(b) the 4:5:6:7:9:11:13 chord might be better approximated by C-E-G-Bb-D-F#-A than by C-
E-G-Bb-D-F#-A, since 11:13 is very close to 3 semitones, and 7:13 is a lot closer to 11
semitones than to 10 semitones.

> As for my "tuning file methodology", I'm still not sure if the basic
> process I use is clear. Can you enlighten me?

I think I understand the rough idea of the process you use, though it seems highly
"nonlinear", if you know what I mean.
. . . Do you mean enlighten you as to my ignorance?

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

/tuning/topicId_18017.html#18017

> I know this issue has been thrashed out repeatedly on this
> list, but I must admit I still don't "get it". If in what
> I would call "5-limit" tuning, B is 15/16 of C, how is that
> not "15-limit" according to your definition?
>
> If a chord includes C,G, and D, of course D is going to be
> tuned as 9/4 of C, but to my mind that doesn't make it "9-limit".
>
> Perhaps it would be simpler just to qualify using the term
> "prime limit" as opposed to "odd limit". Though I am aware of,
> and largely agree with, the argument that "odd limt" more
> accurately reflects the difficulty of tuning a complex dyad
> to JI by ear, when it comes to the actual process of assigning
> tunings to notes, designating a prime limit makes more sense
> to me.

Hi John,

Just thought I'd mention that a big part of the reason I prefer
prime-limit to odd-limit is that prime-limit it the ultimate
notational simplification. Developing an accurate microtonal
notation that still has similarities to regular staff-notation
was a driving force behind a lot of my theoretical work in
the early 1990s; see my paper on "JustMusic notation":

http://www.ixpres.com/interval/monzo/article/article.htm

Once I realized that regular letter-name-and-sharp/flat notation
was based on a (infinite?) 3-limit system, I had the thought
that the simplest way to indicate any other part of the ratio
would be to add the relevant prime factor and its appropriate
exponent to the accidental for that note.

A lot of the inspiration for my notation comes from Ben Johnston's,
which is also based on prime factoring, but I prefer my approach
basing the letter-names on a Pythagorean system to his approach
which uses the 5-limit diatonic major scale as a basis.

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

[I wrote:]
>>I know this issue has been thrashed out repeatedly on this
>>list, but I must admit I still don't "get it". If in what
>>I would call "5-limit" tuning, B is 15/16 of C, how is that
>>not "15-limit" according to your definition?
>>
>>If a chord includes C,G, and D, of course D is going to be
>>tuned as 9/4 of C, but to my mind that doesn't make it "9-limit".
>>
>>Perhaps it would be simpler just to qualify using the term
>>"prime limit" as opposed to "odd limit". Though I am aware of,
>>and largely agree with, the argument that "odd limt" more
>>accurately reflects the difficulty of tuning a complex dyad
>>to JI by ear, when it comes to the actual process of assigning
>>tunings to notes, designating a prime limit makes more sense
>>to me.

[Monz wrote:]
>Just thought I'd mention that a big part of the reason I prefer
>prime-limit to odd-limit is that prime-limit it the ultimate
>notational simplification. Developing an accurate microtonal
>notation that still has similarities to regular staff-notation
>was a driving force behind a lot of my theoretical work in
>the early 1990s; see my paper on "JustMusic notation":
>
>http://www.ixpres.com/interval/monzo/article/article.htm
>
>Once I realized that regular letter-name-and-sharp/flat notation
>was based on a (infinite?) 3-limit system, I had the thought
>that the simplest way to indicate any other part of the ratio
>would be to add the relevant prime factor and its appropriate
>exponent to the accidental for that note.
>
>A lot of the inspiration for my notation comes from Ben Johnston's,
>which is also based on prime factoring, but I prefer my approach
>basing the letter-names on a Pythagorean system to his approach
>which uses the 5-limit diatonic major scale as a basis.

Kyool. The prime limit becomes the number of dimensions in the lattice,
but one can take as many steps along each axis as suits a particular
scale or progression.

Another quibble with "odd-limit" - why are even numbers excluded? It's
an artificial contrivance from beginning to end. Prime limits rule!

JdL

Monz wrote,

>Just thought I'd mention that a big part of the reason I prefer
>prime-limit to odd-limit is that prime-limit it the ultimate
>notational simplification. Developing an accurate microtonal
>notation that still has similarities to regular staff-notation
>was a driving force behind a lot of my theoretical work in
>the early 1990s; see my paper on "JustMusic notation":

Aargh! This hurts! Monz, _please_ look at my definition again. I write that,
for the purposes of describing the resources of (i.e., naming or counting
the notes of) a JI tuning, which is exactly what your notation is about, one
uses the "prime" definition of limit. The "odd" definition of limit is used
for other things entirely -- read Partch's _Genesis of a Music_ to see it in
action. It's not a matter of preference! Sorry for the grumbling . . .

John deLaubenfels:

>Another quibble with "odd-limit" - why are even numbers excluded?

Because in the contexts in which it is used, octave-invariant pitch notation
is assumed -- hence any factors of 2 can be ignored.

>It's
>an artificial contrivance from beginning to end. Prime limits rule!

Oh dear. I have some suggestions for you:

1. Look over the archives where "complexity" was discussed, and read in
particular Dave Keenan's points.

2. Read Harry Partch's _Genesis of a Music_ -- this is where "limit" was
first used -- you'll find three references to "9-limit" in the index.

3. Look over Erv Wilson's work -- he often maps the _same_ pitch to two
different points on a lattice, for different odd factorizations (e.g., 1*9
and 3*3), since there are _different_ consonant relationships with other
notes implied by these representations and his lattice diagrams
symmetrically depict them all . . .

John deLaubenfels wrote,

>It's
>an artificial contrivance from beginning to end. Prime limits rule!

John, even in your own work there is a "limit" to how high an exponent of a
particular prime number you're willing to accomodate. This, of course,
constitutes an odd limit, though in an imprecise sense. So on some level you
clearly understand the importance of the odd limit, yet you refuse to
explicity honor it . . .

Oh yes, one more thing to add to the list -- if you look over the harmonic
entropy posts, you'll see that there were only two measures of complexity
that I found which, when used to "seed" the harmonic entropy calculation,
came out also correctly ranking the entropy values of the lower local minima
of the output. One was n*d, the Tenney measure; the other was the odd-limit,
which is the Partch measure (implied by the One-Footed Bride and other
material in _Genesis of a Music_).

[I wrote:]
>>Another quibble with "odd-limit" - why are even numbers excluded?

[Paul E:]
>Because in the contexts in which it is used, octave-invariant pitch
>notation is assumed -- hence any factors of 2 can be ignored.

>>It's an artificial contrivance from beginning to end. Prime limits
>>rule!

>Oh dear. I have some suggestions for you:

>1. Look over the archives where "complexity" was discussed, and read in
>particular Dave Keenan's points.

>2. Read Harry Partch's _Genesis of a Music_ -- this is where "limit"
>was first used -- you'll find three references to "9-limit" in the
>index.

>3. Look over Erv Wilson's work -- he often maps the _same_ pitch to two
>different points on a lattice, for different odd factorizations (e.g.,
>1*9 and 3*3), since there are _different_ consonant relationships with
>other notes implied by these representations and his lattice diagrams
>symmetrically depict them all . . .

[Paul wrote:]
>John, even in your own work there is a "limit" to how high an exponent
>of a particular prime number you're willing to accomodate. This, of
>course, constitutes an odd limit, though in an imprecise sense. So on
>some level you clearly understand the importance of the odd limit, yet
>you refuse to explicity honor it . . .

>Oh yes, one more thing to add to the list -- if you look over the
>harmonic entropy posts, you'll see that there were only two measures of
>complexity that I found which, when used to "seed" the harmonic entropy
>calculation, came out also correctly ranking the entropy values of the
>lower local minima of the output. One was n*d, the Tenney measure; the
>other was the odd-limit, which is the Partch measure (implied by the
>One-Footed Bride and other material in _Genesis of a Music_).

Paul, I'm sorry, I was being a bit naughty, but I didn't mean to rile
you up so much! Of course I understand the primes aren't the whole
story. Thanks for all the references! More in a moment regarding your
other post on the subject...

JdL

John wrote,
>>Another quibble with "odd-limit" - why are even numbers excluded?

I wrote,
>Because in the contexts in which it is used, octave-invariant pitch
>notation is assumed -- hence any factors of 2 can be ignored.

I might add, yours is just such a context, John.

[I wrote:]
>>>Another quibble with "odd-limit" - why are even numbers excluded?

[Paul E:]
>>Because in the contexts in which it is used, octave-invariant pitch
>>notation is assumed -- hence any factors of 2 can be ignored.

[Paul E again:]
>I might add, yours is just such a context, John.

I was wondering if you'd notice! Quite right, Paul. Again, my
apologies for getting you so riled up!

JdL

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

/tuning/topicId_18017.html#18091

> Monz wrote,
>
> > Just thought I'd mention that a big part of the reason I prefer
> > prime-limit to odd-limit is that prime-limit it the ultimate
> > notational simplification. Developing an accurate microtonal
> > notation that still has similarities to regular staff-notation
> > was a driving force behind a lot of my theoretical work in
> > the early 1990s; see my paper on "JustMusic notation":
>
> Aargh! This hurts! Monz, _please_ look at my definition again.
> I write that, for the purposes of describing the resources of
> (i.e., naming or counting the notes of) a JI tuning, which is
> exactly what your notation is about, one uses the "prime"
> definition of limit. The "odd" definition of limit is used
> for other things entirely -- read Partch's _Genesis of a Music_
> to see it in action. It's not a matter of preference! Sorry
> for the grumbling . . .

Paul, relax... there's no disagreement between us here. I simply
didn't mention odd-limit at all in my post, and was merely giving
John an illustration of one reason I found prime-limit so useful,
and at a particular stage in my own work.

Certainly, I agree with you completely that the use of "odd-limit"
has its place in tuning theory, and an entirely separate basis from
that of "prime".

(OK, so I admit guilt by sin of omission... feel better now?)

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

--- In tuning@y..., MONZ@J... wrote:

/tuning/topicId_18017.html#18154

> --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
>
> /tuning/topicId_18017.html#18091
>
> > Monz wrote,
> >
> > > Just thought I'd mention that a big part of the reason I prefer
> > > prime-limit to odd-limit is that prime-limit it the ultimate
> > > notational simplification. Developing an accurate microtonal
> > > notation that still has similarities to regular staff-notation
> > > was a driving force behind a lot of my theoretical work in
> > > the early 1990s; see my paper on "JustMusic notation":
> >
> > Aargh! This hurts! Monz, _please_ look at my definition again.
> > I write that, for the purposes of describing the resources of
> > (i.e., naming or counting the notes of) a JI tuning, which is
> > exactly what your notation is about, one uses the "prime"
> > definition of limit. The "odd" definition of limit is used
> > for other things entirely -- read Partch's _Genesis of a Music_
> > to see it in action. It's not a matter of preference! Sorry
> > for the grumbling . . .
>
>
>
> Paul, relax... there's no disagreement between us here. I simply
> didn't mention odd-limit at all in my post, and was merely giving
> John an illustration of one reason I found prime-limit so useful,
> and at a particular stage in my own work.
>
> Certainly, I agree with you completely that the use of "odd-limit"
> has its place in tuning theory, and an entirely separate basis from
> that of "prime".
>
> (OK, so I admit guilt by sin of omission... feel better now?)

After a private exchange with Paul, I now understand why he
got so upset about this. I used a bad choice of words in
starting out with "...the reason I prefer prime-limit to
odd-limit...".

It would have been more accurately reflective of the way I
now think about the whole prime-vs.-odd thing if I had written
instead something like "...the reason I generally find prime-
factorization more useful than odd-factorization in my work..."

Given the pages of debate that Paul and I have had in the past
over this issue, and my somewhat grudging acknowledgement that
(as usual) Paul is right, I really should have been more careful
here. Oh well... been very busy, very tired, and very rushed
in scanning thru the Tuning List, lately...

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