Problems with the terms odd/prime limit (was "n EDO is good in the p-limit")

>"As a last note, keep in mind that the 9-odd-limit actually exists as
part of the 7-prime-limit, although nothing's to stop you from talking
about the 5-prime-limit, 9-odd-limit, or the 2.7.9.11-prime-limit, 11
odd limit, etc."

   This goes back in my list of "10000 reasons why I hate the term 'limit' (being ambiguous between 'prime' and 'odd')". So 9 can be reduced to 3 in "prime-limit", but not in odd-limit (IE 9/8 is 9-odd-limit, but 3 prime-limit), correct?

    Misinterpreting statement that say limit without saying "prime" or "odd" seems incurably commonplace on this list (even) and elsewhere because people so often just say "limit" without saying what type of limit.

  What if we coined a terminology such as "9-limit, lowest prime (LP) 3" or even just 9<3>-limit to represent something with 9-odd-limit, 3-prime-limit (or something else along similar lines)?   Then, of course, 7-limit (odd and prime) would become 7<7> limit...and in discussion you could say <7> to infer the prime limit and 7 to infer the odd limit.

    As I understand it...odd-limit is essential and most important anyhow (and thus, I'm thinking, should be noted even if you are only discussing prime limit)...since a dyad's having low prime-limit just generally increases the chance other tones will form lower odd-limit relationships with it, but scientifically guarantees nothing.

On 15 Aug 2011, at 20:18, Michael wrote:

> As I understand it...odd-limit is essential and most important anyhow (and thus, I'm thinking, should be noted even if you are only discussing prime limit)...since a dyad's having low prime-limit just generally increases the chance other tones will form lower odd-limit relationships with it, but scientifically guarantees nothing.

hmmm... I was going to suggest using 7-limit to always mean prime unless the word odd is included..
Why is odd-limit most important?

Steve P.

--- In tuning@yahoogroups.com, Steve Parker <steve@...> wrote:

> hmmm... I was going to suggest using 7-limit to always mean prime unless the word odd is included..
> Why is odd-limit most important?

Prime limit in fact is more important, and it's pretty common for p-limit to default to mean prime limit. If you mean odd limit, it's probably best to say odd limit.

>"Prime limit in fact is more important, and it's pretty common for p-limit to default to mean prime limit"

  Now I am really confused.  Before I have heard Carl, in several cases, say odd-limit is the more important.  Plus I've heard several arguments that virtually every interval can be approximated within a few cents by 5-prime-limit and not even close to that in 5-odd-limit, in an effort to demonstrate how little prime-limit can actually mean (IE 105/64 approximately = 18/11) .  And, yes, I was convinced...very well.

  Now...why is prime limit more important?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Prime limit in fact is more important, and it's pretty common for p-limit to default to mean prime limit"
>
>   Now I am really confused.  Before I have heard Carl, in several cases, say odd-limit is the more important.  Plus I've heard several arguments that virtually every interval can be approximated within a few cents by 5-prime-limit and not even close to that in 5-odd-limit, in an effort to demonstrate how little prime-limit can actually mean (IE 105/64 approximately = 18/11) .  And, yes, I was convinced...very well.

This is obviously true to a mathematician. The infinite ratios of the n-prime-limit for any n >= 3 form what's called a "dense set", which means you can approximate any other interval arbitrarily accurately. You say "within a few cents", but really you could get within a billionth of a cent if you wanted to.

>   Now...why is prime limit more important?

Because if you already have a 3-limit system where you can stack intervals on top of each other and remain in the system, then you can easily get 9/4 just by stacking two 3/2 on top of each other. The key point is that you only need *two* 3/2s to get 9/4 *exactly* right, whereas if you were trying to approximate some new prime, say 5, you can get as close as you want, *but* in order to get closer and closer you have to stack more and more 3/2s until your scale has a ridiculous number of notes in it.

In other words having 3s in your system guarantees you have 9s of low "complexity", where "complexity" basically means the number of notes you have to put in your scale to get a certain interval - the complexity of 9 is just twice the complexity of 3. You can also get arbitrarily accurate 5s or any higher prime, but for a new prime there is always a trade-off between accuracy and complexity.

For odd numbers that are the product of lower primes (9, 15, 21, 25...), you automatically get them exactly right with a finite complexity, so there is no trade-off.

Keenan

Me>"Now...why is prime limit more important?"

Keenan>"Because if you already have a 3-limit system where you can stack
intervals on top of each other and remain in the system, then you can
easily get 9/4 just by stacking two 3/2 on top of each other. The key
point is that you only need *two* 3/2s to get 9/4 *exactly* right,
whereas if you were trying to approximate some new prime, say 5, you can
get as close as you want, *but* in order to get closer and closer you
have to stack more and more 3/2s until your scale has a ridiculous
number of notes in it."

  In other words...it forecasts if you will get a "comma" of error when creating circles of intervals when trying to join them to, say, the octave or an interval of another prime limit (and you will always get an error if the generators you want to intersect within only a handful of iterations/powers have different prime limits), correct?

  And if so, is there an easy way to forecast how much error you will get (IE, between a spiral of 3 prime limit and 5 prime-limit with x number of powers)?

    This all seems to hint prime limit is very important to the formation and comparison of regular temperaments but not consonance,  while odd-limit is more a guide to evaluating consonance but not for forming regular temperaments.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>   In other words...it forecasts if you will get a "comma" of error when creating circles of intervals when trying to join them to, say, the octave or an interval of another prime limit (and you will always get an error if the generators you want to intersect within only a handful of iterations/powers have different prime limits), correct?

Well, you'll always get a comma if there isn't some obvious facorization like 3x3=9. That's the Fundamental Theorem of Arithmetic. The question isn't whether you get a comma, it's where the commas show up and how big they are.

>   And if so, is there an easy way to forecast how much error you will get (IE, between a spiral of 3 prime limit and 5 prime-limit with x number of powers)?

Well, this is basically what the whole regular mapping paradigm is all about, right?

If I understand what you mean when you say "spiral", the 3 prime limit is already in a spiral form even in perfect just intonation, because you have a period 2/1 and a generator 3/2 (or equivalently 4/3). Mathematicians say it's "rank 2". But for 5 prime limit or higher, you have to temper out one or more commas to get it into that form.

And there are convenient tools to find the best ways of doing that.

http://x31eq.com/temper/pregular.html

If you type in 5 for the limit (don't get confused by the instructions - it's really a prime limit...), and something for the error, it will spit out a bunch of systems, many of which have names, of "matching up the different spirals and combining them into one spiral". One of the best for error around 5 cents is meantone, which means the comma 81/80 is tempered out so four stacked approximate 3/2s make an approximate 5/1, as you may already know.

>     This all seems to hint prime limit is very important to the formation and comparison of regular temperaments but not consonance,  while odd-limit is more a guide to evaluating consonance but not for forming regular temperaments.

I agree with this summary.

Keenan

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

>   Now...why is prime limit more important?

Because it's a basic theoretical definition. Odd limit is for octave-equivalent consonance, and is a bit arbitrary. You could have instead Kees limit, or octave-equivalent TE limit, etc etc. I would have thought that this was something you already understood, given your previous comments.

Steve Parker <steve@...> wrote:

> hmmm... I was going to suggest using 7-limit to always
> mean prime unless the word odd is included.. Why is
> odd-limit most important?

The odd limit tells you something about what you're going
to hear. A certain set of intervals are considered to be
consonances, and so will come up a lot in chords, and be
used to connect chords. They might be out of tune but
you'll get some idea what they're going to sound like.

Any interval can be approximated to arbitrary precision by
any two distinct primes. The prime limit tells you nothing
about the music. It describes the algebraic structure
behind the tuning.

In practice, music in a given prime limit will focus on the
simplest ratios within that limit. We formalize this by
using weighted prime limits. If the kind of limit isn't
specified, you can generally assume it's either an odd
limit or a Tenney-weighted prime limit. In practice, it
doesn't usually matter which, because they'll focus on the
same intervals. When you do care about an odd limit it's
best to be explicit about that.

An odd limit uniquely defines a prime limit. A prime limit
suggests an odd limit but sometimes there are
alternatives. For example, the 7-prime limit could be
either the 7-odd or 9-limit (or even a Farey limit or
something else). Odd limits roughly follow Tenney-complexity
cutoffs. You may choose to state an odd-limit because it
means you list a finite set of intervals and musicians will
tend to be familiar with octave equivalence. There's no
cosmic censor preventing you from using intervals outside
the limit you stated. With a temperament, you can even add
intervals from your odd limit to get an approximation of an
interval from a different prime limit.

Graham

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"As a last note, keep in mind that the 9-odd-limit actually exists as
> part of the 7-prime-limit, although nothing's to stop you from talking
> about the 5-prime-limit, 9-odd-limit, or the 2.7.9.11-prime-limit, 11
> odd limit, etc."
>
>    This goes back in my list of "10000 reasons why I hate the term 'limit' (being ambiguous between 'prime' and 'odd')". So 9 can be reduced to 3 in "prime-limit", but not in odd-limit (IE 9/8 is 9-odd-limit, but 3 prime-limit), correct?
>
>     Misinterpreting statement that say limit without saying "prime" or "odd" seems incurably commonplace on this list (even) and elsewhere because people so often just say "limit" without saying what type of limit.
>
>   What if we coined a terminology such as "9-limit, lowest prime (LP) 3" or even just 9<3>-limit to represent something with 9-odd-limit, 3-prime-limit (or something else along similar lines)?   Then, of course, 7-limit (odd and prime) would become 7<7> limit...and in discussion you could say <7> to infer the prime limit and 7 to infer the odd limit.
>
>     As I understand it...odd-limit is essential and most important anyhow (and thus, I'm thinking, should be noted even if you are only discussing prime limit)...since a dyad's having low prime-limit just generally increases the chance other tones will form lower odd-limit relationships with it, but scientifically guarantees nothing.
>

I have been thinking of limit only in terms of the prime limit, like the 5-limit used to form triads, and 7-limit used to form just dominant 7th chords. But as I work with chord tunings, I find that odd-limit determines, visually speaking, how many times an interval can be stacked on each other. For instance a high odd limit with a prime limit of 5 would let you stack just major triads to form extended chords such as the lydian 13th chord using alternating just major and minor 3rds. The kind of dissonance caused by high odd-limit are of a different quality than that caused by high prime limits. An example would be the difference between the 7:9:11 tuning of the augmented triad, and the 16:20:25 (stacked just major 3rds).

Billy

On Mon, Aug 15, 2011 at 3:18 PM, Michael <djtrancendance@...> wrote:
>
> >"As a last note, keep in mind that the 9-odd-limit actually exists as
> part of the 7-prime-limit, although nothing's to stop you from talking
> about the 5-prime-limit, 9-odd-limit, or the 2.7.9.11-prime-limit, 11
> odd limit, etc."
>
>     Misinterpreting statement that say limit without saying "prime" or "odd" seems incurably commonplace on this list (even) and elsewhere because people so often just say "limit" without saying what type of limit.
>
>   What if we coined a terminology such as "9-limit, lowest prime (LP) 3" or even just 9<3>-limit to represent something with 9-odd-limit, 3-prime-limit (or something else along similar lines)?   Then, of course, 7-limit (odd and prime) would become 7<7> limit...and in discussion you could say <7> to infer the prime limit and 7 to infer the odd limit.
>
>     As I understand it...odd-limit is essential and most important anyhow (and thus, I'm thinking, should be noted even if you are only discussing prime limit)...since a dyad's having low prime-limit just generally increases the chance other tones will form lower odd-limit relationships with it, but scientifically guarantees nothing.

9<3> is pretty neat, but I think that prime-limit is the more
important one. To be honest, I don't really care about odd-limit at
all, but some people find it useful as a rule of thumb to tell you how
consonant a chord is. But to my ears, 8:10:12:15 is more consonant
than 8:11:13:14, so I don't use odd-limit much.

The usual argument for odd-limit is that without something like
odd-limit in place, the 5-prime-limit includes dyads like 5^14/3^20,
which is 0.66 cents away from 7/4. Or it could include chords like 1/1
125/64 27/16 65536/50000, or whatever. I just don't see any reason to
care about that possibility, and at any rate there are a lot of
5-prime-limit chords that I like that are better described as being
something like 45-odd-limit, and also plenty of 45-odd-limit chords
outside of the 5-limit that I think sound awful. I suppose if you come
from mainly a common practice paradigm, where major 7 chords sound
"dissonant" and add2 chords sound like suspensions that "need to
resolve," you might find it useful. For anyone out of that paradigm I
don't think it's useful at all.

-Mike

On Tue, Aug 16, 2011 at 5:21 AM, Graham Breed <gbreed@...> wrote:
>
> Steve Parker <steve@...> wrote:
>
> An odd limit uniquely defines a prime limit.

Why couldn't you extend the definition of odd-limit to subgroups of
that prime-limit? I don't see any reason why 11-odd-limit,
4.7.9.11-prime-limit couldn't work, or why 9-odd-limit, 5-prime limit
couldn't work. In fact, I've heard a lot of "new age" type music that
fits into the latter category really well. Wouldn't what I think Gene
called "Kees-limit" generalize odd-limit in such a fashion? Or
whichever one it is that involves drawing hexagons onto the triangular
lattice.

-Mike

On 17 Aug 2011, at 18:25, Mike Battaglia wrote:

> > Steve Parker <steve@pinkrat.co.uk> wrote:
> >
> > An odd limit uniquely defines a prime limit.

I'm not sure who did, but I didn't write this - I don't find odd-limit useful at all for myself.

Steve P.

Sorry, that was supposed to be for Graham.

-Mike

On Wed, Aug 17, 2011 at 1:52 PM, Steve Parker <steve@...> wrote:
>
>
>
> On 17 Aug 2011, at 18:25, Mike Battaglia wrote:
>
> > Steve Parker <steve@...> wrote:
> >
> > An odd limit uniquely defines a prime limit.
>
> I'm not sure who did, but I didn't write this - I don't find odd-limit useful at all for myself.
> Steve P.
>
>

Mike Battaglia <battaglia01@...> wrote:

> Why couldn't you extend the definition of odd-limit to
> subgroups of that prime-limit? I don't see any reason why
> 11-odd-limit, 4.7.9.11-prime-limit couldn't work, or why
> 9-odd-limit, 5-prime limit couldn't work. In fact, I've
> heard a lot of "new age" type music that fits into the
> latter category really well. Wouldn't what I think Gene
> called "Kees-limit" generalize odd-limit in such a
> fashion? Or whichever one it is that involves drawing
> hexagons onto the triangular lattice.

It's an intersection. You'd have the intersection of an
odd limit and a prime limit.

Graham