questions about Paul's "Tuning, Tonality, & 22..."

Ok, some of these may be a little basic, but here goes:

1. Page 1: I'm having touble understanding the term "5 limit"
Does that really mean any fraction made of the factors 2,3,5
in the numerator and denominator? Wouldn't there be an infinite
number of those in the interval (1,2)? so we need some cutoff?

2. On p.2 it says "The major and minor modes gained supremacy
because only in those modes was the diminished 5th disjoint from
the tonic triad". Huh? Isn't the Locrian mode the only one with
the dim 5th in the tonic triad?

3. On p.5, what is the "unaltered diatonic scale"? From footnote 30,
evidently it is not [ 1 9/8/ 5/4 4/3 3/2 5/3 15/8 ]?

More later,
--Jeff

Paul's on the list, and I'll take a stab at the first
two...

>1. Page 1: I'm having touble understanding the term "5 limit"
> Does that really mean any fraction made of the factors 2,3,5
> in the numerator and denominator? Wouldn't there be an infinite
> number of those in the interval (1,2)? so we need some cutoff?

You may be thinking prime-limit. "limit" is odd-limit (it's
Partch's term), where the only thing you're allowed to factor out
is 2.

>2. On p.2 it says "The major and minor modes gained supremacy
> because only in those modes was the diminished 5th disjoint from
> the tonic triad". Huh? Isn't the Locrian mode the only one with
> the dim 5th in the tonic triad?

"Disjoint", not "in".

-Carl

Hi, Carl. Thanks for the reply

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Paul's on the list, and I'll take a stab at the first
> two...
>
> >1. Page 1: I'm having touble understanding the term "5 limit"
> > Does that really mean any fraction made of the factors 2,3,5
> > in the numerator and denominator? Wouldn't there be an infinite
> > number of those in the interval (1,2)? so we need some cutoff?
>
> You may be thinking prime-limit. "limit" is odd-limit (it's
> Partch's term), where the only thing you're allowed to factor out
> is 2.
>
I'm still not sure I understand this concept, or why it is useful.
For example we could form something like 3^40 * 5^22 / 2^78 and
say this is a consonant interval? That seems very wierd...

> >2. On p.2 it says "The major and minor modes gained supremacy
> > because only in those modes was the diminished 5th disjoint from
> > the tonic triad". Huh? Isn't the Locrian mode the only one with
> > the dim 5th in the tonic triad?
>
> "Disjoint", not "in".
>

Am I missing something obvious? Yes, Locrian has a bad tonic triad,
but Ionian and Aeolian are surely not the only ones without the
dim 5th ( Dorian should be ok, etc...)

I guess it all depends on what exactly Paul is defining to be
THE diatonic scale.

--JEff

>> >1. Page 1: I'm having touble understanding the term "5 limit"
>> > Does that really mean any fraction made of the factors 2,3,5
>> > in the numerator and denominator? Wouldn't there be an infinite
>> > number of those in the interval (1,2)? so we need some cutoff?
>>
>> You may be thinking prime-limit. "limit" is odd-limit (it's
>> Partch's term), where the only thing you're allowed to factor out
>> is 2.
>>
>I'm still not sure I understand this concept, or why it is useful.
>For example we could form something like 3^40 * 5^22 / 2^78 and
>say this is a consonant interval? That seems very wierd...

No, that's 5-prime-limit. The complete list of 5-limit ratios is...

6/5 315.6
5/4 386.3
4/3 498.0
3/2 702.0
8/5 813.7
5/3 884.4

...plus any of these altered by factors of 2.

>> >2. On p.2 it says "The major and minor modes gained supremacy
>> > because only in those modes was the diminished 5th disjoint from
>> > the tonic triad". Huh? Isn't the Locrian mode the only one with
>> > the dim 5th in the tonic triad?
>>
>> "Disjoint", not "in".
>>
>
>Am I missing something obvious? Yes, Locrian has a bad tonic triad,
>but Ionian and Aeolian are surely not the only ones without the
>dim 5th ( Dorian should be ok, etc...)
>
>I guess it all depends on what exactly Paul is defining to be
>THE diatonic scale.

Disjoint here means something like adjacent. With no sharps or
flats in the key signature, the dim 5th occurs between b and f,
adjacent to Amin and CMaj, the tonic triads of the two dominant
modes. In Paul's model "characteristic dissonances" like the
dim 5th aid in position-finding, but need to resolve, making
adjacent/disjoint consonances special.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Disjoint here means something like adjacent.

It means 'having no notes in common with'.

> With no sharps or
> flats in the key signature, the dim 5th occurs between b and f,
> adjacent to Amin and CMaj, the tonic triads of the two dominant
> modes.

These are the only two diatonic triads that don't contain *either* b
*or* f.

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:

> 3. On p.5, what is the "unaltered diatonic scale"? From footnote 30,
> evidently it is not [ 1 9/8/ 5/4 4/3 3/2 5/3 15/8 ]?

no, it's simply the 'natural' diatonic scale, as opposed to (for
example) the harmonic minor scale, which is an *altered* diatonic
scale.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
>
> > 3. On p.5, what is the "unaltered diatonic scale"? From footnote
30,
> > evidently it is not [ 1 9/8/ 5/4 4/3 3/2 5/3 15/8 ]?
>
> no, it's simply the 'natural' diatonic scale, as opposed to (for
> example) the harmonic minor scale, which is an *altered* diatonic
> scale.

I'm sorry to be dense, but this point is the main thing holding
me up in the paper. If the natural diatonic scale is not the
set of fractions I quoted above, then what exactly *is* it?

Is it the Pythagorean fractions [1 9/8 81/64 4/3 3/2 27/16 243/128] ?

--Jeff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> Ok, some of these may be a little basic, but here goes:
>
> 1. Page 1: I'm having touble understanding the term "5 limit"
> Does that really mean any fraction made of the factors 2,3,5
> in the numerator and denominator?

No, that would have been the Hindemith definition, which I explicitly
reject in the second sentence of my paper. Rather than the prime
factors, my definition (which agrees with Partch's original usage of
the term) concerns the *numbers* themselves, except for factors of 2.

> Wouldn't there be an infinite
> number of those in the interval (1,2)?

No, these are the only ways to make ratios out of odd numbers (we
start with odd because factors of 2 are treated separately) no higher
than 5:

1/1
3/1
5/1
1/3
1/5
5/3
3/5

That's it. Now we can freely insert factors of 2, and the only
resulting ratios that end up between 1/1 and 2/1, inclusive, are

1/1
6/5
5/4
4/3
3/2
8/5
5/3
2/1

A quite finite list!

> 2. On p.2 it says "The major and minor modes gained supremacy
> because only in those modes was the diminished 5th disjoint from
> the tonic triad". Huh? Isn't the Locrian mode the only one with
> the dim 5th in the tonic triad?

Yes, but in the Dorian, Phrygian, Lydian, and Mixolydian modes, the
diminished 5th shares one note with the tonic triad, so is not
disjoint from it. Rather than a resolution by stepwise contrary
motion from the diminished 5th to notes of the tonic triad, you have
at best a resolution by oblique motion, which is not as strong.
Resolution by stepwise contrary motion has provided the framework for
harmonic progression since the middle ages, when such resolutions
were from then-dissonant seconds, thirds, sixths, and sevenths to
then-consonant fourths, fifths or unisons, fourths or octaves, and
fifths, respectively. In the Renaissance, thirds and sixths were used
as (and tuned as) consonances, so many of these resolutions lost
their power; by allowing the diminished 5th as a vertical interval, a
new dissonance-resolving-stepwise-through-contrary-motion-to-a-
consonance pattern became possible, and in turn facilitated the
estabishment of tonality, largely through its clear position-finding
implications.

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
> wrote:
> >
> > > 3. On p.5, what is the "unaltered diatonic scale"? From
footnote
> 30,
> > > evidently it is not [ 1 9/8/ 5/4 4/3 3/2 5/3 15/8 ]?
> >
> > no, it's simply the 'natural' diatonic scale, as opposed to (for
> > example) the harmonic minor scale, which is an *altered* diatonic
> > scale.
>
> I'm sorry to be dense, but this point is the main thing holding
> me up in the paper. If the natural diatonic scale is not the
> set of fractions I quoted above, then what exactly *is* it?

The white keys on a piano, for example. The exact tuning system used
is largely irrelevant -- see my other paper, _The Forms Of Tonality_,
for more information about what precisely is special about the
unaltered diatonic scale, regardless of what tuning system is used to
render it.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
> > Ok, some of these may be a little basic, but here goes:
> >
> > 1. Page 1: I'm having touble understanding the term "5 limit"
> > Does that really mean any fraction made of the factors 2,3,5
> > in the numerator and denominator?
>
> No, that would have been the Hindemith definition, which I
explicitly
> reject in the second sentence of my paper.

It's not the correct definition for consonance, obviously. It *is*
the correct defintion for the totality of 5-limit ratios, because
otherwise we do not have the property of closure. While this is an
ideal rather than an actual property (a piano, for instance, has 88
keys, not an infinite number) it makes things very much simpler.

>It *is* the correct defintion for the totality of 5-limit ratios,
>because otherwise we do not have the property of closure.

No, it is not. Maybe Partch didn't care about closure.

-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
> > Ok, some of these may be a little basic, but here goes:
> >
> > 1. Page 1: I'm having touble understanding the term "5 limit"
> > Does that really mean any fraction made of the factors 2,3,5
> > in the numerator and denominator?
>
> No, that would have been the Hindemith definition, which I
explicitly
> reject in the second sentence of my paper. Rather than the prime
> factors, my definition (which agrees with Partch's original usage
of
> the term) concerns the *numbers* themselves, except for factors of
2.
>
> > Wouldn't there be an infinite
> > number of those in the interval (1,2)?
>
> No, these are the only ways to make ratios out of odd numbers (we
> start with odd because factors of 2 are treated separately) no
higher
> than 5:
>
> 1/1
> 3/1
> 5/1
> 1/3
> 1/5
> 5/3
> 3/5
>
> That's it. Now we can freely insert factors of 2, and the only
> resulting ratios that end up between 1/1 and 2/1, inclusive, are
>
> 1/1
> 6/5
> 5/4
> 4/3
> 3/2
> 8/5
> 5/3
> 2/1
>
> A quite finite list!

This may sound silly, and if does, I apologise in advance, but I have
always considered suspicious the treatment x-limit-ratio approach
gives to ratios as 11/7 and 77/64.

If we accept octave-equivalence, why should those sounds be treated
differently, if both mean the presence of the 7th and 11th harmonics?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >It *is* the correct defintion for the totality of 5-limit ratios,
> >because otherwise we do not have the property of closure.
>
> No, it is not. Maybe Partch didn't care about closure.

What was Partch's definition, and what is right about it? Closure
makes things so much easier it seems foolish to dismiss it like this.

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

> This may sound silly, and if does, I apologise in advance, but I
have
> always considered suspicious the treatment x-limit-ratio approach
> gives to ratios as 11/7 and 77/64.
>
> If we accept octave-equivalence, why should those sounds be treated
> differently, if both mean the presence of the 7th and 11th
>harmonics?

I have no time for a full answer right now, just a tease: Any time we
hear a timbre with 12 or more harmonic overtones, we're hearing the
interval 11:7 right there in every single note, but of course we're
not hearing 77:64. For a different point of view, consider the
coincidence of harmonics when two periodic tones are played at the
same time -- first at the interval of 11:7, then at the interval of
77:64. Which has more coincidences?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >It *is* the correct defintion for the totality of 5-limit ratios,
> > >because otherwise we do not have the property of closure.
> >
> > No, it is not. Maybe Partch didn't care about closure.
>
> What was Partch's definition, and what is right about it? Closure
> makes things so much easier it seems foolish to dismiss it like
>this.

What are you guys arguing about, and why? Clearly there are two
different limit definitions, as Monz's dictionary will tell you. The
prime-limit definition is obviously useless for consonance, because
any conceivable interval can be approximated with arbitrary accuracy
with a 5-prime-limit ratio. Since your own constructs have been
completely in line with my ideas and Partch's here, Gene, I'm having
trouble understanding the point of this latest post of yours.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
>
> > This may sound silly, and if does, I apologise in advance, but I
> have
> > always considered suspicious the treatment x-limit-ratio approach
> > gives to ratios as 11/7 and 77/64.
> >
> > If we accept octave-equivalence, why should those sounds be
treated
> > differently, if both mean the presence of the 7th and 11th
> >harmonics?
>
> I have no time for a full answer right now, just a tease: Any time
we
> hear a timbre with 12 or more harmonic overtones, we're hearing the
> interval 11:7 right there in every single note, but of course we're
> not hearing 77:64. For a different point of view, consider the
> coincidence of harmonics when two periodic tones are played at the
> same time -- first at the interval of 11:7, then at the interval of
> 77:64. Which has more coincidences?

Now for another hint, using octave-equivalence this time:

Let's take the intervals which occur among the first 6 harmonics, or
which have the property that at least one of the first 6 harmonics of
the first note coincide with at least one of the first 6 harmonics of
the other note:

1:1
2:1
3:1
3:2
4:1
4:3
5:1
5:2
5:3
5:4
6:1
6:5

Now for each of these intervals, form the series of its octave
inversions and extensions (that is, the series of intervals belonging
to the same interval-class under octave equivalence of pitches --
meaning the series of intervals that can be derived from the original
interval by transposing one or both pitches by any number of octaves):

1:1 - 1:1, 2:1, 4:1, 8:1, 16:1, 32:1, 64:1, 128:1, 256:1 . . .
2:1 - 1:1, 2:1, 4:1, 8:1, 16:1, 32:1, 64:1, 128:1, 256:1 . . .
3:1 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
3:2 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
4:1 - 1:1, 2:1, 4:1, 8:1, 16:1, 32:1, 64:1, 128:1, 256:1 . . .
4:3 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
5:1 - 32:5, 16:5, 8:5, 5:4, 5:2, 5:1, 10:1, 20:1, 40:1 . . .
5:2 - 32:5, 16:5, 8:5, 5:4, 5:2, 5:1, 10:1, 20:1, 40:1 . . .
5:3 - 48:5, 24:5, 12:5, 6:5, 5:3, 10:3, 20:3, 40:3 . . .
5:4 - 32:5, 16:5, 8:5, 5:4, 5:2, 5:1, 10:1, 20:1, 40:1 . . .
6:1 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
6:5 - 48:5, 24:5, 12:5, 6:5, 5:3, 10:3, 20:3, 40:3 . . .

Notice that the number 15 does not appear in either the numerator or
denominator of any of these ratios, despite that fact that you might
have said that such a ratio (with 15) would only "mean the presence
of the 3rd and 5th harmonics". The same is true of 9, 25, 27, 45, or
any other odd number higher than 5, despite the fact that the *prime*
factors of these numbers may all be 5 or lower.

Hopefully this helps clarify this in the context of my paper and your
question, but I'd be more than happy to continue this discussion --
I've only been wrestling with these issues for about 15 years!

-Paul

>> >It *is* the correct defintion for the totality of 5-limit ratios,
>> >because otherwise we do not have the property of closure.
>>
>> No, it is not. Maybe Partch didn't care about closure.
>
>What was Partch's definition, and what is right about it? Closure
>makes things so much easier it seems foolish to dismiss it like this.

Both Paul and I just gave that def, and you yourself said it
makes sense acoustically. It's about the simplest octave-equivalent
version of Tenney height one can concoct.

You mentioned something about infinity. I thought closure meant
that every element was connected to some other element via some
operation. Clearly there are an infinite number of both n-limit
and n-prime-limit ratios.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >1. Page 1: I'm having touble understanding the term "5 limit"
> >> > Does that really mean any fraction made of the factors 2,3,5
> >> > in the numerator and denominator? Wouldn't there be an
infinite
> >> > number of those in the interval (1,2)? so we need some
cutoff?
> >>
> >> You may be thinking prime-limit. "limit" is odd-limit (it's
> >> Partch's term), where the only thing you're allowed to factor out
> >> is 2.
> >>
> >I'm still not sure I understand this concept, or why it is useful.
> >For example we could form something like 3^40 * 5^22 / 2^78 and
> >say this is a consonant interval? That seems very wierd...
>
> No, that's 5-prime-limit. The complete list of 5-limit ratios is...
>
> 6/5 315.6
> 5/4 386.3
> 4/3 498.0
> 3/2 702.0
> 8/5 813.7
> 5/3 884.4
>
> ...plus any of these altered by factors of 2.

i think as of now we should always qualify "limit" with
either "prime-" or "odd-", whichever is applicable.

it would entirely eliminate the continued confusion
over these two concepts.

-monz

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> >
> > --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >
> > > > It *is* the correct defintion for the totality of
> > > > 5-limit ratios, because otherwise we do not have
> > > > the property of closure.
> > >
> > > No, it is not. Maybe Partch didn't care about closure.
> >
> > What was Partch's definition, and what is right about it?
> > Closure makes things so much easier it seems foolish to
> > dismiss it like this.
>
> What are you guys arguing about, and why? Clearly there
> are two different limit definitions, as Monz's dictionary
> will tell you. The prime-limit definition is obviously
> useless for consonance, because any conceivable interval
> can be approximated with arbitrary accuracy with a
> 5-prime-limit ratio. Since your own constructs have been
> completely in line with my ideas and Partch's here, Gene,
> I'm having trouble understanding the point of this latest
> post of yours.

according to a recent addition to the Tuning Dictionary
with which paul helped me

http://sonic-arts.org/dict/ratio-of.htm

the hopefully-standard current term for "the totality
of 5-odd-limit ratios" is "ratios of 5".

i think some of the confusion has to do with the fact
that some theorists, including (notably here in the case
of the Tuning Dictionary as a reference for list-members)
myself, ascribe much significance to the *prime*-limits
observable in sound-complexes, even in regard to the subject
of consonance (or more properly, concordance).

Gene, i think this is what paul is referring to. he,
Graham, Dave Keenan and a lot of other "heavyweights"
around here ascribe significance to the *odd*-limits
in regarding concordance. (and i see their point and
am inclined to agree to some extent ... but see above ...)

as i said, from now on let's always use "odd-limit"
or "prime-limit" accordingly ... and sprinkle in
"ratios of ..." (with odd-limit understood) liberally.

;-)

-monz

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> What are you guys arguing about, and why? Clearly there are two
> different limit definitions, as Monz's dictionary will tell you.

This is what I said, and what Carl denied.

The
> prime-limit definition is obviously useless for consonance, because
> any conceivable interval can be approximated with arbitrary
accuracy
> with a 5-prime-limit ratio. Since your own constructs have been
> completely in line with my ideas and Partch's here, Gene, I'm
having
> trouble understanding the point of this latest post of yours.

I simply said 5-limit consonances and 5-limit intervals were two
different things, and that 5-limit intervals had the property of
closure. Carl said Partch did not agree.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Both Paul and I just gave that def, and you yourself said it
> makes sense acoustically. It's about the simplest octave-equivalent
> version of Tenney height one can concoct.

Apparently you are talking about something completely different than
I am, though I don't know what.

> You mentioned something about infinity. I thought closure meant
> that every element was connected to some other element via some
> operation. Clearly there are an infinite number of both n-limit
> and n-prime-limit ratios.

What are n-limit ratios and n-prime-limit ratios? Is n a prime number?

Closure under multiplication means that is you have two elements in
the set, their product is in the set. Closure under inversion means
if q is in the set, 1/q is in the set. If you have both kinds of
closure, the set constitutes a group, which is convenient.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> according to a recent addition to the Tuning Dictionary
> with which paul helped me
>
> http://sonic-arts.org/dict/ratio-of.htm
>
> the hopefully-standard current term for "the totality
> of 5-odd-limit ratios" is "ratios of 5".

I'm afraid this isn't true; 4/3 is a 5-odd-limit ratio, but not a
ratio of 5 using your definition.

> Gene, i think this is what paul is referring to. he,
> Graham, Dave Keenan and a lot of other "heavyweights"
> around here ascribe significance to the *odd*-limits
> in regarding concordance.

Since I was not talking about concordances this isn't relevant to
anything I said.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> Now for another hint, using octave-equivalence this time:
>
> Let's take the intervals which occur among the first 6 harmonics,
or
> which have the property that at least one of the first 6 harmonics
of
> the first note coincide with at least one of the first 6 harmonics
of
> the other note:
>
> 1:1
> 2:1
> 3:1
> 3:2
> 4:1
> 4:3
> 5:1
> 5:2
> 5:3
> 5:4
> 6:1
> 6:5
>
> Now for each of these intervals, form the series of its octave
> inversions and extensions (that is, the series of intervals
belonging
> to the same interval-class under octave equivalence of pitches --
> meaning the series of intervals that can be derived from the
original
> interval by transposing one or both pitches by any number of
octaves):
>
> 1:1 - 1:1, 2:1, 4:1, 8:1, 16:1, 32:1, 64:1, 128:1, 256:1 . . .
> 2:1 - 1:1, 2:1, 4:1, 8:1, 16:1, 32:1, 64:1, 128:1, 256:1 . . .
> 3:1 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
> 3:2 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
> 4:1 - 1:1, 2:1, 4:1, 8:1, 16:1, 32:1, 64:1, 128:1, 256:1 . . .
> 4:3 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
> 5:1 - 32:5, 16:5, 8:5, 5:4, 5:2, 5:1, 10:1, 20:1, 40:1 . . .
> 5:2 - 32:5, 16:5, 8:5, 5:4, 5:2, 5:1, 10:1, 20:1, 40:1 . . .
> 5:3 - 48:5, 24:5, 12:5, 6:5, 5:3, 10:3, 20:3, 40:3 . . .
> 5:4 - 32:5, 16:5, 8:5, 5:4, 5:2, 5:1, 10:1, 20:1, 40:1 . . .
> 6:1 - 32:3, 16:3, 8:3, 4:3, 3:2, 3:1, 6:1, 12:1, 24:1, 48:1 . . .
> 6:5 - 48:5, 24:5, 12:5, 6:5, 5:3, 10:3, 20:3, 40:3 . . .
>
> Notice that the number 15 does not appear in either the numerator
or
> denominator of any of these ratios, despite that fact that you
might
> have said that such a ratio (with 15) would only "mean the presence
> of the 3rd and 5th harmonics". The same is true of 9, 25, 27, 45,
or
> any other odd number higher than 5, despite the fact that the
*prime*
> factors of these numbers may all be 5 or lower.
>
> Hopefully this helps clarify this in the context of my paper and
your
> question, but I'd be more than happy to continue this discussion --
> I've only been wrestling with these issues for about 15 years!
>
> -Paul

Well, it's a disparate discussion, for I'm well down in the learning
curve, and these debates likely teach more to me than to anybody else.

Yep, I see your point, Paul. I was thinking of a tri-triadic diatonic
scale, in which notes are formed going down or up certain intervals
based on harmonics: the "third" (5/4) and the "fifth" (3/2). So with
C as fundamental of the scale, B is formed going up a fifth and then
a third, and A equals to a third up and a fifth down (please octave-
rectify). And both notes represent the same function in two different
triads (third of dominant and subdominant chords). But you're deadly
right, in a consonantic way, A is more familiar to C and shares with
that note many more overtones.

Max.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > according to a recent addition to the Tuning Dictionary
> > with which paul helped me
> >
> > http://sonic-arts.org/dict/ratio-of.htm
> >
> > the hopefully-standard current term for "the totality
> > of 5-odd-limit ratios" is "ratios of 5".
>
> I'm afraid this isn't true; 4/3 is a 5-odd-limit ratio, but not a
> ratio of 5 using your definition.

Yes, Gene is correct, and Monz was incorrect in the above. I'm sure
this was only a temporary slip on Monz's part, as I've gone over this
with him before.

>according to a recent addition to the Tuning Dictionary
>with which paul helped me
>
>http://sonic-arts.org/dict/ratio-of.htm
>
>the hopefully-standard current term for "the totality
>of 5-odd-limit ratios" is "ratios of 5".

No, that isn't right. Read your pages again! 5-limit
ratios include 3-limit ratios, but "ratios of 5" do not.

-Carl

>> What are you guys arguing about, and why? Clearly there are two
>> different limit definitions, as Monz's dictionary will tell you.
>
>This is what I said, and what Carl denied.

You said "limit", and if you don't specify "odd-limit" or
"prime-limit", "limit" means "odd-limit". We've (esp. Paul)
worked very hard on this point for many years on this list.

>I simply said 5-limit consonances and 5-limit intervals were two
>different things, and that 5-limit intervals had the property of
>closure. Carl said Partch did not agree.

"5-limit intervals" is confusing terminology outside of a
paragraph like this. You should say 5-prime-limit.

-Carl

>> Both Paul and I just gave that def, and you yourself said it
>> makes sense acoustically. It's about the simplest octave-equivalent
>> version of Tenney height one can concoct.
>
>Apparently you are talking about something completely different than
>I am, though I don't know what.
>
>> You mentioned something about infinity. I thought closure meant
>> that every element was connected to some other element via some
>> operation. Clearly there are an infinite number of both n-limit
>> and n-prime-limit ratios.
>
>What are n-limit ratios and n-prime-limit ratios? Is n a prime number?

n is just a number. n-prime-limit ratios are all rational numbers
whose factorization contains terms no greater than n. n-limit ratios
are all rational numbers which, after factors of 2 are removed,
contain no term greater than n.

>Closure under multiplication means that is you have two elements in
>the set, their product is in the set.

I think that's the same as what I said.

>Closure under inversion means
>if q is in the set, 1/q is in the set. If you have both kinds of
>closure, the set constitutes a group, which is convenient.

Both odd- and prime- limits are closed under inversion, but only
prime- is closed under multiplication. Convenience must bow to
pyschoacoustics, and both must bow to Partch.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What are you guys arguing about, and why? Clearly there are two
> >> different limit definitions, as Monz's dictionary will tell you.
> >
> >This is what I said, and what Carl denied.
>
> You said "limit", and if you don't specify "odd-limit" or
> "prime-limit", "limit" means "odd-limit". We've (esp. Paul)
> worked very hard on this point for many years on this list.

It seems to me the prime limit is more important, and unless odd
limit is specified, prime limit should be the default assumption.
Obviously, however, if I say "intervals" I am talking about prime
limits, and if I say "consonances" I am talking about odd limits, and
no further qualifications should be necessary.

> >I simply said 5-limit consonances and 5-limit intervals were two
> >different things, and that 5-limit intervals had the property of
> >closure. Carl said Partch did not agree.
>
> "5-limit intervals" is confusing terminology outside of a
> paragraph like this. You should say 5-prime-limit.

There is nothing whatever confusing about "5-limit intervals" if you
take note of the "intevals"; you imported the idea that consonances
had something to do with it, but I did not say so.

Anyway, I've been talking like this for a long time now. How did it
suddenly become an issue?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> n is just a number. n-prime-limit ratios are all rational numbers
> whose factorization contains terms no greater than n.

It probably won't matter to most people, but number theorists almost
never use "n" to mean a number we specifically know to be prime. A
lower case "p" or a script "l" is most common.

> Both odd- and prime- limits are closed under inversion, but only
> prime- is closed under multiplication. Convenience must bow to
> pyschoacoustics, and both must bow to Partch.

This is a manifestly absurd statement.

>It seems to me the prime limit is more important, and unless odd
>limit is specified, prime limit should be the default assumption.

I can see why you'd think that, but it isn't the case in my
experience.

>Obviously, however, if I say "intervals" I am talking about prime
>limits, and if I say "consonances" I am talking about odd limits, and
>no further qualifications should be necessary.

"Intervals", in the common usage, usually means odd-limit.
This isn't tuning-math.

>> >I simply said 5-limit consonances and 5-limit intervals were two
>> >different things, and that 5-limit intervals had the property of
>> >closure. Carl said Partch did not agree.
>>
>> "5-limit intervals" is confusing terminology outside of a
>> paragraph like this. You should say 5-prime-limit.
>
>There is nothing whatever confusing about "5-limit intervals" if you
>take note of the "intevals"; you imported the idea that consonances
>had something to do with it, but I did not say so.

In your imaginary world. But in the real world "intervals" does
not carry this specialized meaning.

>Anyway, I've been talking like this for a long time now. How did it
>suddenly become an issue?

Because suddenly, we're not on tuning-math.

-Carl

>> n is just a number. n-prime-limit ratios are all rational numbers
>> whose factorization contains terms no greater than n.
>
>It probably won't matter to most people, but number theorists almost
>never use "n" to mean a number we specifically know to be prime. A
>lower case "p" or a script "l" is most common.

We don't know n is prime. It's any number.

>> Both odd- and prime- limits are closed under inversion, but only
>> prime- is closed under multiplication. Convenience must bow to
>> pyschoacoustics, and both must bow to Partch.
>
>This is a manifestly absurd statement.

No, it isn't, and I can and will defend it against anything you
think you've cooked up.

-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > according to a recent addition to the Tuning Dictionary
> > > with which paul helped me
> > >
> > > http://sonic-arts.org/dict/ratio-of.htm
> > >
> > > the hopefully-standard current term for "the totality
> > > of 5-odd-limit ratios" is "ratios of 5".
> >
> > I'm afraid this isn't true; 4/3 is a 5-odd-limit ratio, but not a
> > ratio of 5 using your definition.
>
> Yes, Gene is correct, and Monz was incorrect in the above.
> I'm sure this was only a temporary slip on Monz's part,
> as I've gone over this with him before.

oops, yep. sorry, my bad.

that's what happens when i stay up all night ...

speaking of which, i hope someone has found a way to
get my javascript applet working.

-monz

on 11/29/03 11:27 AM, Carl Lumma <ekin@lumma.org> wrote:

>>> What are you guys arguing about, and why? Clearly there are two
>>> different limit definitions, as Monz's dictionary will tell you.
>>
>> This is what I said, and what Carl denied.
>
> You said "limit", and if you don't specify "odd-limit" or
> "prime-limit", "limit" means "odd-limit". We've (esp. Paul)
> worked very hard on this point for many years on this list.

You've been working hard, but ever since I've been on this list, I've been
reading Monz's dictionary, and monz's dictionary on this page:

http://sonic-arts.org/dict/limit.htm

states:

> limit
>
> 1. prime limit: A pitch system in Just Intonation where all ratios are of
> integers containing no prime factors higher than n is said to be an "n-limit"
> system.
>
> [from Paul Erlich, private communication]

so someone please straighten this out because you are confusing newbies! ;)

And regarding this problem:

on 11/29/03 12:12 PM, Carl Lumma <ekin@lumma.org> wrote:

> We don't know n is prime. It's any number.

the above definition handles that case, although it allows that 11-limit is
the same as 12-limit, which is indeed strange.

-Kurt

> http://sonic-arts.org/dict/limit.htm
>
>states:
>
>> limit
>>
>> 1. prime limit: A pitch system in Just Intonation where all ratios
>> are of integers containing no prime factors higher than n is said
>> to be an "n-limit" system.
>>
>> [from Paul Erlich, private communication]
>
>so someone please straighten this out because you are confusing
>newbies! ;)

This page gives definitions for both prime- and odd- limit. It
doesn't show limit alone as defaulting to odd, and hence it does
reflect usage among the 'American gamelan' JI crowd (Lou Harrison,
David Doty, etc.), for who limit alone defaults to prime.. But
Partch coined the term and Partch's limit was odd, and on these
lists where you see limit alone most often what's meant is odd.

>the above definition handles that case, although it allows that
>11-limit is the same as 12-limit, which is indeed strange.

It is unorthodox for n to be even for either odd- or prime-
limits, but perfectly acceptable within the context of a concise
definition for Gene. There is also the so-called whole-number
limit, and normally if you see "12-limit" it would refer to that.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What are you guys arguing about, and why? Clearly there are two
> >> different limit definitions, as Monz's dictionary will tell you.
> >
> >This is what I said, and what Carl denied.
>
> You said "limit", and if you don't specify "odd-limit" or
> "prime-limit", "limit" means "odd-limit". We've (esp. Paul)
> worked very hard on this point for many years on this list.

Carl, I think the original Carl Lumma died and an impostor has taken
his place. :)

It seems that everyone except a very tiny fraction of us have ignored
that Partch based his limit definition on his "ratios of" concept,
which is of course the "odd-limit" concept. Partch does begin to
speak of the "extended 5-limit", etc., with ratios he
considered "dissonant" -- not worthy of any further differentiation
into consonant and dissonant categories, as his One-Footed Bride
shows -- and just about everyone after Partch has made the leap to
a "closure"-satisfying "prime-limit" concept.

Both concepts have been enormously useful in my own theories

-- the odd-limit concept in octave-equivalent consonance theory,
evaluation of temperament errors, and odd lattices; chord
construction (e.g., ASSes); and to a limited extent (including
deriving and understanding concepts like Partch Diamonds, Eikosanies,
Wilson CPS scales and their stellations), scale construction

-- the prime-limit concept in most lattices, through the Tenney
lattice to octave-specific consonance theory, and eminently in scale
construction through the Fokker Periodicity Block concept.

Which brings to mind a question for Gene: Are there any ETs where
the 'best' val uses different intervals for 3:1 and 9:3, where 'best'
is defined by *my* odd-limit rms error calculation?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> n-prime-limit ratios are all rational numbers
> >> whose factorization contains terms no greater than n.
> >
> >It probably won't matter to most people, but number theorists
almost
> >never use "n" to mean a number we specifically know to be prime. A
> >lower case "p" or a script "l" is most common.
>
> We don't know n is prime. It's any number.

Give me an example of n-prime-limit where n isn't prime.

on 11/29/03 7:31 PM, Carl Lumma <ekin@lumma.org> wrote:

>> http://sonic-arts.org/dict/limit.htm
>>
>> states:
>>
>>> limit
>>>
>>> 1. prime limit: A pitch system in Just Intonation where all ratios
>>> are of integers containing no prime factors higher than n is said
>>> to be an "n-limit" system.
>>>
>>> [from Paul Erlich, private communication]
>>
>> so someone please straighten this out because you are confusing
>> newbies! ;)
>
> This page gives definitions for both prime- and odd- limit. It
> doesn't show limit alone as defaulting to odd,

Yes, I see - sorry for the confusion. Reading only the prime limit in a
quick scan my eye caught "n-limit" and assume that implied that n-limit
therefore implied prime limit. But I see "n-limit" occurs in the odd-limit
definition also.

> and hence it does
> reflect usage among the 'American gamelan' JI crowd (Lou Harrison,
> David Doty, etc.), for who limit alone defaults to prime.. But
> Partch coined the term and Partch's limit was odd, and on these
> lists where you see limit alone most often what's meant is odd.

However, this is good info, and the "limit" page might well benefit from the
addition of a little history of usage, as well as any "default"
interpretation that may apply for a given context, and also mentioning
odd-limit as being a kind of "global default" if indeed that is true. This
is pretty essential terminology and it would be good to have it covered
well.

-Kurt

Speaking of being email crippled, ignore the subject that appeared in my
last message, since it is absolutely meaningless. I happenned to click
reply and then realized I wanted to send directly to you.

-Kurt

on 11/29/03 10:05 PM, Kurt Bigler <kkb@breathsense.com> wrote:

> Speaking of being email crippled, ignore the subject that appeared in my
> last message, since it is absolutely meaningless. I happenned to click
> reply and then realized I wanted to send directly to you.

Speaking of being email-crippled. Ignore that last message of mine, which
was not intended to be to the list at all, and is entirely meaningless in
this context.

-Kurt

>> and hence it does
>> reflect usage among the 'American gamelan' JI crowd (Lou Harrison,
>> David Doty, etc.), for who limit alone defaults to prime.. But
>> Partch coined the term and Partch's limit was odd, and on these
>> lists where you see limit alone most often what's meant is odd.

>However, this is good info, and the "limit" page might well benefit from
>the addition of a little history of usage, as well as any "default"
>interpretation that may apply for a given context, and also mentioning
>odd-limit as being a kind of "global default" if indeed that is true.

As I say I think it's true here to some extent, and it seems a good
direction to shoot for. On tuning-math the context is a little more
subtle, but it's there and you'll see me use "limit" to mean "prime
limit" there all over the place.

The Lou Harrison contingent also has a valid 'context' of sorts, as
I've taken pains to point out many times in the past: the prime limit
of a *piece* of music often reflects the odd limit of the vertical
consonances it employs.

>This is pretty essential terminology and it would be good to have it
>covered well.

Well in defense of monz's dictionary, 1. and 2. are

You know, I've been thinking. The tuning dictionary would be awesome
on a Wiki, so everybody could contribute changes. It could get huge
beyond what monz or anyone could be expected to do. And we could
leave the contributions signed, or, go with the Wiki Way of anonymity --
it's supposed to reduce flame wars and force people to be objective,
and for content to stand on its own.

And then I think, Wikipedia's already got a tie-in for us. And
they have the software already running. We can all go over there and
kick some ass, and come back and discuss it here if we like (even ask
for comments before updating or changing a page). And our work would
get tremendous exposure.

What do you think? Especially you, monz, since it's your dictionary.
Well, it is to say you are someone who could make a huge contribution
to a WikiTuningDictionary, and/or state your policy on others' copying
of tuning dictionary content into Wikipedia. And of course both sites
can (and should) run concurrently.

-Carl

Speaking of being e-mail crippled...

>>as well as any "default"
>>interpretation that may apply for a given context
//
>>This is pretty essential terminology and it would be good to have it
>>covered well.
>
>Well in defense of monz's dictionary, 1. and 2. are

...

Make that, "

In defense of monz' dictionary, '1. and 2.' is a standard sort
of way dictionaries say 'depends on context'. And if it depends
on contxet, the reader is supposed to be able to figure it out.
Maybe an example of the contexts...

"... 15-limit diamond ..."
"Common-practice Western music conforms to the 5-limit"

...Are examples of odd-limit context. And these are examples
of prime-limit...

"Let us represent 7-limit notes by means of what I've taken to
calling "monzos"--row vectors of integers representing the exponents
of 2, 3, 5, and 7 respectively."

"(1) There is a "sharp" axis, c, c sharp, etc., where if q is a 5-
limit rational number, then h5(q) (where h5=[5, 8, 12] is the 5-limit
standard val for the 5-et) counts the number of steps along this axis
of the "sharp" coordinate. This axis is inclined at an angle of
76.102 degrees upward."

"... 5-limit commas ..."

"

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >There is nothing whatever confusing about "5-limit intervals" if
you
> >take note of the "intevals"; you imported the idea that
consonances
> >had something to do with it, but I did not say so.
>
> In your imaginary world. But in the real world "intervals" does
> not carry this specialized meaning.

Sorry, but it is only in your specialized world that all intervals
are assumed to be consonant. The rest of the world makes no such
assumption.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> You've been working hard, but ever since I've been on this list,
I've been
> reading Monz's dictionary, and monz's dictionary on this page:
>
> http://sonic-arts.org/dict/limit.htm
>
> states:
>
> > limit
> >
> > 1. prime limit: A pitch system in Just Intonation where all
ratios are of
> > integers containing no prime factors higher than n is said to be
an "n-limit"
> > system.

That seems right, but I'd like it even better if "n" was a prime and
called "p" instead. I see no use for 8-limit systems being a synonym
for 7-limit systems, and even less for pi-limit systems meaning the
same as 3-limit systems.

> > We don't know n is prime. It's any number.
>
>
> the above definition handles that case, although it allows that 11-
limit is
> the same as 12-limit, which is indeed strange.

It's worse than strange, it is confusing and counterproductive. It
should be changed.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> Which brings to mind a question for Gene: Are there any ETs where
> the 'best' val uses different intervals for 3:1 and 9:3,
where 'best'
> is defined by *my* odd-limit rms error calculation?

I would expect so; we could certainly look.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_48658.html#48813

> on 11/29/03 7:31 PM, Carl Lumma <ekin@l...> wrote:
>
> >> http://sonic-arts.org/dict/limit.htm
> >>
> >> states:
> >>
> >>> limit
> >>>
> >>> 1. prime limit: A pitch system in Just Intonation where all
ratios
> >>> are of integers containing no prime factors higher than n is
said
> >>> to be an "n-limit" system.
> >>>
> >>> [from Paul Erlich, private communication]
> >>
> >> so someone please straighten this out because you are confusing
> >> newbies! ;)
> >
> > This page gives definitions for both prime- and odd- limit. It
> > doesn't show limit alone as defaulting to odd,
>
> Yes, I see - sorry for the confusion. Reading only the prime limit
in a
> quick scan my eye caught "n-limit" and assume that implied that n-
limit
> therefore implied prime limit. But I see "n-limit" occurs in the
odd-limit
> definition also.
>
> > and hence it does
> > reflect usage among the 'American gamelan' JI crowd (Lou Harrison,
> > David Doty, etc.), for who limit alone defaults to prime.. But
> > Partch coined the term and Partch's limit was odd, and on these
> > lists where you see limit alone most often what's meant is odd.
>
> However, this is good info, and the "limit" page might well benefit
from the
> addition of a little history of usage, as well as any "default"
> interpretation that may apply for a given context, and also
mentioning
> odd-limit as being a kind of "global default" if indeed that is
true. This
> is pretty essential terminology and it would be good to have it
covered
> well.
>
> -Kurt

***Well, I'm embarassed to say that now I'm completely confused about
this. Could somebody please summarize in maybe three sentences and
maybe about 10 numbers or so the difference between *prime* and *odd*
limit? My "understanding" is that *prime* subsumes the lower prime
limits under it, wheras *odd* includes ratios that might be outside
that paradigm??

Thanks!

J. Pehrson

>Sorry, but it is only in your specialized world that all intervals
>are assumed to be consonant. The rest of the world makes no such
>assumption.

I'm not claiming "intervals" conforms to odd-limit. An interval is
any distance, not necessarily a tonespace approximation. They are
also sometimes scale distances.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Sorry, but it is only in your specialized world that all intervals
> >are assumed to be consonant. The rest of the world makes no such
> >assumption.
>
> I'm not claiming "intervals" conforms to odd-limit. An interval is
> any distance, not necessarily a tonespace approximation. They are
> also sometimes scale distances.

You are assumming "5-limit interval" can be any real number?

on 11/30/03 2:18 PM, Joseph Pehrson <jpehrson@rcn.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

[stuff regarding prime-limit versus odd-limit]

> ***Well, I'm embarassed to say that now I'm completely confused about
> this. Could somebody please summarize in maybe three sentences and
> maybe about 10 numbers or so the difference between *prime* and *odd*
> limit? My "understanding" is that *prime* subsumes the lower prime
> limits under it, wheras *odd* includes ratios that might be outside
> that paradigm??

Actually prime-limit includes ratios that odd-limit does not (for a given
number). For example 3-prime-limit includes ratios involving 9, but
3-odd-limit does not. So odd-limit is more restrictive, and for myself,
more generally useful as a result. With prime limit you would not be
distinguishing 9-limit from 11-limit, first because 9 is not prime. 9
versus 11 limit is a very useful distinction, so odd-limit is to me very
useful. If I say 7 limit and that includes 49, that is less useful to me
beause those higher-numbered ratios are "leaking in" to the picture. But
that's just my point of view.

-Kurt

> Thanks!
>
> J. Pehrson

> You are assumming "5-limit interval" can be any real number?

No. I don't know how we misunderstood eachother. I thought
you were claiming "5-limit interval" should mean "5-prime-limit
interval" by default, and I am claiming that usage does not
support this.

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_48658.html#48842

>>
> Actually prime-limit includes ratios that odd-limit does not (for a
given
> number). For example 3-prime-limit includes ratios involving 9, but
> 3-odd-limit does not. So odd-limit is more restrictive, and for
myself,
> more generally useful as a result.

***Thanks, Kurt. I believe I was reversing the two in my mind
somehow...

Joseph

try the updated version now.

i decided to keep "n-limit" because i didn't like
the look of "o-limit" for the odd-limit definition.
(confusion with zero, etc.)

if there's enough consensus that i should change
the descriptions in the definitions to "p-limit"
and "o-limit", then i will.

-monz

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> > You've been working hard, but ever since I've been on this list,
> I've been
> > reading Monz's dictionary, and monz's dictionary on this page:
> >
> > http://sonic-arts.org/dict/limit.htm
> >
> > states:
> >
> > > limit
> > >
> > > 1. prime limit: A pitch system in Just Intonation where all
> ratios are of
> > > integers containing no prime factors higher than n is said to
be
> an "n-limit"
> > > system.
>
> That seems right, but I'd like it even better if "n" was a prime
and
> called "p" instead. I see no use for 8-limit systems being a
synonym
> for 7-limit systems, and even less for pi-limit systems meaning the
> same as 3-limit systems.
>
> > > We don't know n is prime. It's any number.
> >
> >
> > the above definition handles that case, although it allows that
11-
> limit is
> > the same as 12-limit, which is indeed strange.
>
> It's worse than strange, it is confusing and counterproductive. It
> should be changed.

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > You are assumming "5-limit interval" can be any real number?
>
> No. I don't know how we misunderstood eachother. I thought
> you were claiming "5-limit interval" should mean "5-prime-limit
> interval" by default, and I am claiming that usage does not
> support this.

I know of no meaning for "5-limit interval" other than "5-prime-limit
interval". Note I say "interval", not "consonance".

>try the updated version now.
>
>i decided to keep "n-limit" because i didn't like
>the look of "o-limit" for the odd-limit definition.
>(confusion with zero, etc.)
>
>if there's enough consensus that i should change
>the descriptions in the definitions to "p-limit"
>and "o-limit", then i will.

http://sonic-arts.org/dict/limit.htm

That looks good, except I'd take out "Just Intonation"
under 1.. We often use things like "7-limit" to mean
approximations that are not "audibly indistinguishable"
from JI.

-Carl

>> > You are assumming "5-limit interval" can be any real number?
>>
>> No. I don't know how we misunderstood eachother. I thought
>> you were claiming "5-limit interval" should mean "5-prime-limit
>> interval" by default, and I am claiming that usage does not
>> support this.
>
>I know of no meaning for "5-limit interval" other than "5-prime-
limit
>interval". Note I say "interval", not "consonance".

Well, take some time to re-read this thread and you might learn
something.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >I know of no meaning for "5-limit interval" other than "5-prime-
> limit
> >interval". Note I say "interval", not "consonance".
>
> Well, take some time to re-read this thread and you might learn
> something.

That is less than helpful. Do you mean something other than either
"5-prime-limit interval" (redundent) or "5-odd-limit consonance"
(pleonastic) and if so, what?

>> >I know of no meaning for "5-limit interval" other than "5-prime-
>> limit
>> >interval". Note I say "interval", not "consonance".
>>
>> Well, take some time to re-read this thread and you might learn
>> something.
>
>That is less than helpful.

You're right, and sorry for being an ass. I'll take it down a
notch.

>Do you mean something other than either
>"5-prime-limit interval" (redundent) or "5-odd-limit consonance"
>(pleonastic) and if so, what?

I mean, if you look over the tuning list archives, you should
see instances of "5-limit intervals" where 5-odd-limit is meant.

And if context sets it up, "5-limit intervals" could even mean
scale intervals, like I was trying to say... 'in the diatonic
scale, the "5-limit intervals" are 4ths, 5ths, 3rds and 6ths'.

on 11/30/03 1:20 AM, Carl Lumma <ekin@lumma.org> wrote:

> Speaking of being e-mail crippled...
>
>>> as well as any "default"
>>> interpretation that may apply for a given context
> //
>>> This is pretty essential terminology and it would be good to have it
>>> covered well.
>>
>> Well in defense of monz's dictionary, 1. and 2. are
>
> ...
>
> Make that, "
>
> In defense of monz' dictionary,

monz' dictionary needs no defense. I do consider monz' dictionary to be a
part of the public wealth, and in that sense I take fractional
responsibility for its potential, including possible improvements.

> '1. and 2.' is a standard sort
> of way dictionaries say 'depends on context'.
> And if it depends
> on contxet, the reader is supposed to be able to figure it out.

and of course I was misreading it anyway (sorry again)

> Maybe an example of the contexts...
>
> "... 15-limit diamond ..."
> "Common-practice Western music conforms to the 5-limit"
>
> ...Are examples of odd-limit context. And these are examples
> of prime-limit...
>
> "Let us represent 7-limit notes by means of what I've taken to
> calling "monzos"--row vectors of integers representing the exponents
> of 2, 3, 5, and 7 respectively."
>
> "(1) There is a "sharp" axis, c, c sharp, etc., where if q is a 5-
> limit rational number, then h5(q) (where h5=[5, 8, 12] is the 5-limit
> standard val for the 5-et) counts the number of steps along this axis
> of the "sharp" coordinate. This axis is inclined at an angle of
> 76.102 degrees upward."
>
> "... 5-limit commas ..."

well that does help, actually, just to know explicitly that the N-limit
construct will regularly be used without qualification and might mean either
prime or odd. The implications of the analogy to a "normal" dictionary
might have been more obvious to me if the term being defined were "N-limit"
rather than just "limit" as a category for separate definitions of
prime-limit and odd-limit, which is what I took it to be. Mind you what I
took it to be is not any indication of how a normal person would take it.
;)

-Kurt

>
> "
>
> -Carl

on 11/30/03 8:20 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
>>> I know of no meaning for "5-limit interval" other than "5-prime-
>> limit
>>> interval". Note I say "interval", not "consonance".
>>
>> Well, take some time to re-read this thread and you might learn
>> something.
>
> That is less than helpful. Do you mean something other than either
> "5-prime-limit interval" (redundent) or "5-odd-limit consonance"
> (pleonastic) and if so, what?

In support of Gene here, none of the discussion in this thread involved any
clarification of possible contextual cues given by the presence of the word
"interval", so rereading it would not add anything.

Such cues would be very counter-intuitive to me however. From the outside
it looks like all of this is about intervals. So "limit" is about
intervals, whether odd or prime. I just want to point out the non-arcane
side of this. If the presence of the word "interval" or "consonance" is
truly contextually relevant, then it would be good to have that noted in a
place where everyone using this language will be referring to it. One of
the best places for that to happpen (at least for starters) is the monz
tuning dictionary.

-Kurt

on 11/30/03 6:12 PM, monz <monz@attglobal.net> wrote:

> try the updated version now.
>
> i decided to keep "n-limit" because i didn't like
> the look of "o-limit" for the odd-limit definition.
> (confusion with zero, etc.)
>
> if there's enough consensus that i should change
> the descriptions in the definitions to "p-limit"
> and "o-limit", then i will.

I don't think so. The "form" being defined is actually the linguistic form

n-limit

The issue is that the form n-limit is actually ambiguous, and using p versus
o for the variable name in different contexts would serve to obscure rather
than clarifying the fact of this ambiguity.

Personally I would like to see something like 11-olimit or 11-plimit come
into use. If we can have utones and otones then why not olimits and
plimits? But the history of language can not be changed, and so except in
special cases language can not be changed readily by design.

N in n-limit should be perhaps be in italic to indicate that it is a
variable. At least that is one common use for italics in metalinguistic
contexts. The limit page is inconsistent on this admittedly minor point.

The "Concordance" link on the limit page is broken. Or maybe monz is
working on this as we speak.

-Kurt

>
>
> -monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I mean, if you look over the tuning list archives, you should
> see instances of "5-limit intervals" where 5-odd-limit is meant.

I don't know what that means, if it doesn't mean 5-limit consonances.

> And if context sets it up, "5-limit intervals" could even mean
> scale intervals, like I was trying to say... 'in the diatonic
> scale, the "5-limit intervals" are 4ths, 5ths, 3rds and 6ths'.

These are, of course, consonances, not intervals in any general sense.
If you mean consonances, it seems to me you should say so.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>If the presence of the word "interval" or "consonance" is
> truly contextually relevant, then it would be good to have that
noted in a
> place where everyone using this language will be referring to it.

I think it highly relevant. I take "5-odd-limit-interval" to be a so
far undefined term, unless it is intended as a synonym for
5-odd-limit-consonance, or 5-odd-limit consonant interval. If so,
then it's a confusing one and I recommend no one use it.

>> I mean, if you look over the tuning list archives, you should
>> see instances of "5-limit intervals" where 5-odd-limit is meant.
>
>I don't know what that means, if it doesn't mean 5-limit consonances.

It means 5-limit consonances if to you consonances are exactly
given by odd-limit.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> It means 5-limit consonances if to you consonances are exactly
> given by odd-limit.

This is equally true of any other list of consonant intervals--that
the set of such intervals is whatever it is. The point is, it is
proposed as being such a set.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > Which brings to mind a question for Gene: Are there any ETs where
> > the 'best' val uses different intervals for 3:1 and 9:3,
> where 'best'
> > is defined by *my* odd-limit rms error calculation?
>
> I would expect so; we could certainly look.

I think that's a good idea -- we shouldn't be so "prime-focused" when
defining vals or anything else.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> With prime limit you would not be
> distinguishing 9-limit from 11-limit, first because 9 is not
prime. 9
> versus 11 limit is a very useful distinction, so odd-limit is to me
very
> useful.

Kurt, I think you may be confusing something . . . don't you mean 9-
limit vs. 7-limit, not 9-limit vs. 11-limit? It's the 9-limit and the
7-limit that both get treated the same (as 7-limit) under a prime-
limit definition.

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >try the updated version now.
> >
> >i decided to keep "n-limit" because i didn't like
> >the look of "o-limit" for the odd-limit definition.
> >(confusion with zero, etc.)
> >
> >if there's enough consensus that i should change
> >the descriptions in the definitions to "p-limit"
> >and "o-limit", then i will.
>
> http://sonic-arts.org/dict/limit.htm
>
> That looks good, except I'd take out "Just Intonation"
> under 1.. We often use things like "7-limit" to mean
> approximations that are not "audibly indistinguishable"
> from JI.
>
> -Carl

How about "A pitch system in Just Intonation, or an approximation
thereto, where . . ."

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> > That looks good, except I'd take out "Just Intonation"
> > under 1.. We often use things like "7-limit" to mean
> > approximations that are not "audibly indistinguishable"
> > from JI.

> How about "A pitch system in Just Intonation, or an approximation
> thereto, where . . ."

I don't think audible distinguishibility is the point; a meantone
triad is audibly distinct from a JI triad, but still a 5-limit
entity. My point of view has always been that something is p-limit if
there is a homomorphic mapping from the JI p-limit, but putting that
in Joe's dictionary would probably cause some people to choke.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
> > > That looks good, except I'd take out "Just Intonation"
> > > under 1.. We often use things like "7-limit" to mean
> > > approximations that are not "audibly indistinguishable"
> > > from JI.
>
> > How about "A pitch system in Just Intonation, or an approximation
> > thereto, where . . ."
>
> I don't think audible distinguishibility is the point; a meantone
> triad is audibly distinct from a JI triad, but still a 5-limit
> entity.

Exactly why I said "approximation" and not "audibly indistinguishable
approximation" above.

>How about "A pitch system in Just Intonation, or an approximation
>thereto, where . . ."

'dat sounds fine.

-C.

At 02:12 PM 12/1/2003, you wrote:
>--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
>> > That looks good, except I'd take out "Just Intonation"
>> > under 1.. We often use things like "7-limit" to mean
>> > approximations that are not "audibly indistinguishable"
>> > from JI.
>
>> How about "A pitch system in Just Intonation, or an approximation
>> thereto, where . . ."
>
>I don't think audible distinguishibility is the point; a meantone
>triad is audibly distinct from a JI triad, but still a 5-limit
>entity.

"audibly indistinguishable" comes from the Just Intonation link.
Don't tell me you want to get back into the defining JI thing...
:)

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> At 02:12 PM 12/1/2003, you wrote:
> >--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> >> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >
> >> > That looks good, except I'd take out "Just Intonation"
> >> > under 1.. We often use things like "7-limit" to mean
> >> > approximations that are not "audibly indistinguishable"
> >> > from JI.
> >
> >> How about "A pitch system in Just Intonation, or an
approximation
> >> thereto, where . . ."
> >
> >I don't think audible distinguishibility is the point; a meantone
> >triad is audibly distinct from a JI triad, but still a 5-limit
> >entity.
>
> "audibly indistinguishable" comes from the Just Intonation link.
> Don't tell me you want to get back into the defining JI thing...
> :)
>
> -Carl

Carl, we're constantly referring to meantone, and even pelogic, as "5-
limit temperaments" on tuning-math. I have no idea what you're
getting at with the "Just Intonation link" . . .

>> >> > That looks good, except I'd take out "Just Intonation"
>> >> > under 1.. We often use things like "7-limit" to mean
>> >> > approximations that are not "audibly indistinguishable"
>> >> > from JI.
>> >
>> >> How about "A pitch system in Just Intonation, or an
>approximation
>> >> thereto, where . . ."
>> >
>> >I don't think audible distinguishibility is the point; a meantone
>> >triad is audibly distinct from a JI triad, but still a 5-limit
>> >entity.
>>
>> "audibly indistinguishable" comes from the Just Intonation link.
>> Don't tell me you want to get back into the defining JI thing...
>> :)
>>
>> -Carl
>
>Carl, we're constantly referring to meantone, and even pelogic, as
>"5-limit temperaments" on tuning-math.

That's why I pointed it out, and you fixed it!!!!!!!!!!!!!!!!!!!!!!

>I have no idea what you're
>getting at with the "Just Intonation link" . . .

The term "Just Intonation" in the above definition is a link to
the definition of JI, which in turn contains "audibly
indistinguishable".

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> > That looks good, except I'd take out "Just Intonation"
> >> >> > under 1.. We often use things like "7-limit" to mean
> >> >> > approximations that are not "audibly indistinguishable"
> >> >> > from JI.
> >> >
> >> >> How about "A pitch system in Just Intonation, or an
> >approximation
> >> >> thereto, where . . ."
> >> >
> >> >I don't think audible distinguishibility is the point; a
meantone
> >> >triad is audibly distinct from a JI triad, but still a 5-limit
> >> >entity.
> >>
> >> "audibly indistinguishable" comes from the Just Intonation link.
> >> Don't tell me you want to get back into the defining JI thing...
> >> :)
> >>
> >> -Carl
> >
> >Carl, we're constantly referring to meantone, and even pelogic, as
> >"5-limit temperaments" on tuning-math.
>
> That's why I pointed it out, and you fixed it!!!!!!!!!!!!!!!!!!!!!!

Somehow, probably because of Gene's post, I missed the word "not" in
your original statement! So sorry, Carl
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

>> >Carl, we're constantly referring to meantone, and even pelogic, as
>> >"5-limit temperaments" on tuning-math.
>>
>> That's why I pointed it out, and you fixed it!!!!!!!!!!!!!!!!!!!!!!
>
>Somehow, probably because of Gene's post, I missed the word "not" in
>your original statement! So sorry, Carl

Whew. No sweat.

-C.

on 12/1/03 1:58 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> With prime limit you would not be
>> distinguishing 9-limit from 11-limit, first because 9 is not
> prime. 9
>> versus 11 limit is a very useful distinction, so odd-limit is to me
> very
>> useful.
>
> Kurt, I think you may be confusing something . . . don't you mean 9-
> limit vs. 7-limit, not 9-limit vs. 11-limit? It's the 9-limit and the
> 7-limit that both get treated the same (as 7-limit) under a prime-
> limit definition.

Here I was not referring to the fact that prime limit doesn't make a certain
distinction, but rather that a speaker/writer would not be likely to be
making the stated distinction. Rather than 9 vs 11 they would be
distinguishing 7 vs 11, first because 9 is not prime, and second because as
a result 9 is synonymous with 7, with 7 being to the point and 9 being
obscurative.

The other point was that in prime limit there is no way to make a 9 versus
11 distinction whereas with odd limit there is. And 9 versus 11 limit is a
particularly important distinction for me. So yes, that is actually what I
meant to say, but I did not manage to be clear enough.

-Kurt

on 12/1/03 2:22 PM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>>> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>>
>>>> That looks good, except I'd take out "Just Intonation"
>>>> under 1.. We often use things like "7-limit" to mean
>>>> approximations that are not "audibly indistinguishable"
>>>> from JI.
>>
>>> How about "A pitch system in Just Intonation, or an approximation
>>> thereto, where . . ."
>>
>> I don't think audible distinguishibility is the point; a meantone
>> triad is audibly distinct from a JI triad, but still a 5-limit
>> entity.
>
> Exactly why I said "approximation" and not "audibly indistinguishable
> approximation" above.

Hmm. I'm having the hunch that the real point is somewhere in-between. To
be a "reasonable approximation" in spite of an audible distinction, you want
the listener to be able to "suspend the distinction" (as in "suspension of
judgement"). This might refer to some sense of JI "function" so that you
could say "JI-functionally equivalent to". This might depend on some theory
(e.g. harmonic entropy) to objectify. Yet ultimately it might be
subjective. The question is whether "approximation" is well-enough defined.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/1/03 1:58 PM, Paul Erlich <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> >> With prime limit you would not be
> >> distinguishing 9-limit from 11-limit, first because 9 is not
> > prime. 9
> >> versus 11 limit is a very useful distinction, so odd-limit is to
me
> > very
> >> useful.
> >
> > Kurt, I think you may be confusing something . . . don't you mean
9-
> > limit vs. 7-limit, not 9-limit vs. 11-limit? It's the 9-limit and
the
> > 7-limit that both get treated the same (as 7-limit) under a prime-
> > limit definition.
>
> Here I was not referring to the fact that prime limit doesn't make
a certain
> distinction, but rather that a speaker/writer would not be likely
to be
> making the stated distinction. Rather than 9 vs 11 they would be
> distinguishing 7 vs 11, first because 9 is not prime, and second
because as
> a result 9 is synonymous with 7, with 7 being to the point and 9
being
> obscurative.

You lost me, Kurt. In what event would a speaker note be likely to be
making the stated distinction?

> The other point was that in prime limit there is no way to make a 9
versus
> 11 distinction whereas with odd limit there is. And 9 versus 11
limit is a
> particularly important distinction for me. So yes, that is
actually what I
> meant to say, but I did not manage to be clear enough.

Well maybe, as Gene might be trying to say, if you're speaking
precisely, there's always a *context* clear enough to decide which
limit definition is warranted. For one thing (and I'm writing for
Joseph now too) prime limit gives an infinity of ratios in every
crevice between any two given ratios -- while an odd limit gives a
set with a finite number of notes in the octave.

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> on 11/30/03 6:12 PM, monz <monz@a...> wrote:
>
> > try the updated version now.
> >
> > i decided to keep "n-limit" because i didn't like
> > the look of "o-limit" for the odd-limit definition.
> > (confusion with zero, etc.)
> >
> > if there's enough consensus that i should change
> > the descriptions in the definitions to "p-limit"
> > and "o-limit", then i will.
>
> I don't think so. The "form" being defined is actually
> the linguistic form
>
> n-limit
>
> The issue is that the form n-limit is actually ambiguous,
> and using p versus o for the variable name in different
> contexts would serve to obscure rather than clarifying
> the fact of this ambiguity.

right ... this is another reason why i wanted to leave it
as "n-limit" instead of changing it to the less-generic
"p-limit" and "o-limit". thanks for pointing it out.
always need to keep definitions generic.

> Personally I would like to see something like 11-olimit
> or 11-plimit come into use. If we can have utones and
> otones then why not olimits and plimits? But the history
> of language can not be changed, and so except in special
> cases language can not be changed readily by design.

don't be too quick to decide that. one of the reasons
i decided to compile the dictionary was to help out those
who are new to tuning and having trouble wading thru
the sea of strange terminology ... but another reason
was that i saw the opportunity to shape the form of
future discourse to some degree by deciding myself
(or with the help of colleagues) on definite "standard"
terms and procedures.

> N in n-limit should be perhaps be in italic to indicate
> that it is a variable. At least that is one common use
> for italics in metalinguistic contexts. The limit page
> is inconsistent on this admittedly minor point.

well, you were certainly right about that. thanks to
your post, i saw that i had used the italic-n in definition 1
(prime-limit) but used a plain-n in definition 2 (odd-limit).
i've made it consistently italic now. thanks.

> The "Concordance" link on the limit page is broken. Or
> maybe monz is working on this as we speak.

i simply spelled the webpage filename incorrectly.
it's fixed now. thanks.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> right ... this is another reason why i wanted to leave it
> as "n-limit" instead of changing it to the less-generic
> "p-limit" and "o-limit".

Why are those less generic??

> well, you were certainly right about that. thanks to
> your post, i saw that i had used the italic-n in definition 1
> (prime-limit) but used a plain-n in definition 2 (odd-limit).
> i've made it consistently italic now. thanks.

Monz, I think *both* definitions need to be modified to allow for the
fact that we're often referring to these ratios' approximations,
rather than strictly their just-intonation ideals, when we use the
term 'limit'.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > right ... this is another reason why i wanted to leave it
> > as "n-limit" instead of changing it to the less-generic
> > "p-limit" and "o-limit".
>
> Why are those less generic??
>
> > well, you were certainly right about that. thanks to
> > your post, i saw that i had used the italic-n in definition 1
> > (prime-limit) but used a plain-n in definition 2 (odd-limit).
> > i've made it consistently italic now. thanks.
>
> Monz, I think *both* definitions need to be modified to
> allow for the fact that we're often referring to these
> ratios' approximations, rather than strictly their
> just-intonation ideals, when we use the term 'limit'.

yep, that sounds right ... and i saw your other recent
posts on this subject.

but it's late and i'm too tired to do it now ... if you'd
like to make an addition to the definitions i'll be happy
to add them in tomorrow! ;-)

-monz

on 12/2/03 12:31 AM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
>> right ... this is another reason why i wanted to leave it
>> as "n-limit" instead of changing it to the less-generic
>> "p-limit" and "o-limit".
>
> Why are those less generic??

Let me give this a try. Formally they are no less generic. For anyone who
knows that the free variable is a place holder and has a mind that always
works in pure mathematical forms, p-limit will read formally identical to
o-limit. But for those less experienced in metalinguistic conventions using
different names p and o will obscure the common form. In this case I really
want this page to be a definition of <n>-limit, and I'd almost like to see
the page renamed to that, though I recognize this creates other problems.
(Here I'm using angle brackets <> in place of italics, as is a convention in
some contexts.) As I see it, items "1" and "2" on the page are alternate
definitions of <n>-limit. Using a different name for the free variable in
the two alternate definitions does not not fit with making the page a
definition of <n>-limit.

Also, using p versus o does not in any way correspond to a restriction in
the values taken on the the free variable, for example, as i,j,k are in some
circles used for integers while x,y,z are used for real numbers (etc.). The
definitions given on the limits page require neither p to be prime nor o to
be odd (although common usage would have them be so).

I'm only talking about this definition page. Where do you see advantage for
using "p-limit" and "o-limit"? Neither the p nor the o appears when the
variable is no longer a place-holder, in actual language when an arabic
integer appears in its place. "11" looks the same when replacing "p" as
when replacing "o". "11-p-limit" on the other hand would be an example of a
useful construct. Then we could talk about <n>-p-limit versus <n>-o-limit,
and have separate definitions for p-limit and o-limit (with no italics). Or
just use <n>-plimit and <n>-olimit without the extra hyphen, as I previously
suggested.

(Keep in mind I have never been witness to any tuning-math discussions.)

-Kurt

This reply turned into a late-night ramble, which still might
interest someone. Sorry for the tangentiality of it . . .

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Hmm. I'm having the hunch that the real point is somewhere in-
between. To
> be a "reasonable approximation" in spite of an audible distinction,
you want
> the listener to be able to "suspend the distinction" (as
in "suspension of
> judgement"). This might refer to some sense of JI "function" so
that you
> could say "JI-functionally equivalent to". This might depend on
some theory
> (e.g. harmonic entropy) to objectify.

Yes, this would be the approach for the odd-limit definition, or for
the prime-limit definition if one is solely requiring the consonances
to be "reasonably approximated", and harmonic entropy has a free
parameter which is largely dependent on timbre . . .

However, even if the consonances are reasonably approximated
according to harmonic entropy appropriate to your timbre, some of the
ratios within the prime limit may end up mapped to intervals quite
distant from their JI renditions. For example, in meantone 81:80 is
mapped to the unison (0 cents), and hence so is any *power* of 81:80,
no matter how large! I *still* think there is a sort of functional
equivalence, in that a progression in JI that manages to traverse
several commas will still be heard as ending in the same key it
began, only 'detuned' in a rather disturbing way . . . same with
other, 'non-vanishing' intervals.

What I am suggesting is that if we have 'odd'-functional equivalence,
which implies a certain upper bound on the cents errors on the
consonances, we can then automatically use the same tuning system to
effect 'prime'-functional equivalence, up through the highest 'prime-
limit' not exceeding the 'odd-limit' of the consonances, as
the 'prime-limit' ratios other than the consonances are really only
understood, even in JI, as the result of stacking, or progressing
through, several consonances, and have no acoustical 'specialness'
that is disturbed when their sizes in cents are different from the JI
values.

Hey Gene, I think I've demonstrated a case where it's actually useful
to say 'odd-limit' and 'prime-limit'!

> Yet ultimately it might be
> subjective. The question is whether "approximation" is well-enough
> defined.

Mathematically, any *mapping* can be defined as your "approximation",
even if it makes no sense acoustically. And what makes sense
acoustically can be quite controversial. On the tuning-math list,
we've been talking primarily about temperament. Temperament is based
on taking the infinite lattice of JI, usually of a certain *prime
limit* (but in general defined in terms of *any* basis of consonant
intervals, whether nonconsecutive primes or even ratios that don't
include a fundamental), and lessening the number of dimensions of
infinitude by eliminating certain "unison vectors" or commas, either
purely algebraically/abstractly, or by adjusting the original
consonances slightly so that the commas "vanish". Monz is about to
release some wonderful software that will show this in action and
allow you to make music with the resulting tunings.

The classic example of this is meantone, which is based on the 5-
limit lattice with 81:80 (the syntonic comma) vanishing; thus the
number of dimensions of infinitude of this 5-limit system is reduced
from 2 to 1. That and one other type of temperament, which is based
on the 7-limit lattice with 64:63 and 50:49 vanishing (thus reducing
it from 3 infinite dimensions to 1) -- as well as the notion
of 'chromatic unison vectors', which partition the infinitude into a
finite 'unaltered scale' and its occurences in chromatically altered
form ('sharp', 'double sharp', . . . 'flat', 'double flat' . . .) --
are the subject of this paper:

http://www.lumma.org/tuning/erlich/erlich-tFoT.pdf

(Note that I was still using the *prime* definition of limit in the
above paragraph!)

Anyway, there is a wide range of useful (to different people,
perhaps) error levels to consider. Besides meantone, the most famous
5-limit temperament with one dimension of infinitude is schismic
(32768:32805, the 2-cent 'schisma', tempered out among eight 3:2s and
one 5:4). Smaller and smaller commas require you to go further and
further out in the lattice to find them. Clearly, though, the greater
the error you will allow, the more *simple commas* you will consider
tempering out, and simple commas lead to *simple temperaments* where
the infinite dimensions are not only made finite, but *small* and
manageable. If you are using 'cloudy' timbres with weak and/or
inharmonic partials, you can get away with more error, and find
temperaments like 'pelogic' (135:128 vanishing), where the period is
the octave and the generator is around 523 cents, hence its 'native'
scales resemble the 5- and 7-note 'pelog' scales. This sort of thing,
with the novel chord progressions it allows, is more interesting than
hearing the same old just-intontion sounds in this or that system
that happens to be slightly more manageable.

With some thought, you can see that if the comma you're tempering out
is n:d, then the 'complexity' of the temperament it defines -- the
length in the lattice of the 'finite dimension' it introduces -- is
roughly proportional to log(n), or just as well, log(d). Now the size
of the comma in cents, since it is small, will be roughly
proportional to |n-d|/d, or just as well, |n-d|/n. Hence if we temper
out the comma, the average error it will introduce into the 'rungs'
of the lattice is roughly proportional to the size of the comma in
cents divided by the 'complexity'.

It would be nice to make a list of commas, at least 5-limit ones
(prime-limit of course), where some 'badness' function of average
error and complexity is not too great. If we want an infinite list,
finite within each complexity range, we can use what Gene calls 'log-
flat badness' and get a list like this:

/tuning/database?
method=reportRows&tbl=10&sortBy=3

We didn't actually compile an infinite list; rather, we cut off the
low end by error and the high end by complexity, so after you click
on 'Next' enough times, you'll see the most complex temperament on
the list, based on the 'atom of kirnberger' as its "vanishing comma".

If we want a finite list, Gene tells us that Baker's Theorem (I wish
I knew more) assures us that we can use a 'badness' function
called 'epimericity', which is

log(|n-d|)/log(d), or just as well, log(|n-d|)/log(n)

This can be expressed as

log("avg. error")/"complexity" + log("complexity")/"complexity" + 1

I just spent a couple of hours scratching my head -- this appears at
first to be a *decreasing* function of complexity -- but it isn't
since log("avg. error") will almost always be negative!

Anyway, if we toss out epimericities above 0.5, and 'commas' larger
than 600 cents, here's the list of 5-limit commas, epimericities, and
names of temperaments (from the database above)

16:15 0 father
6:5 0 ------
81:80 0 meantone
4:3 0 ------
9:8 0 ------
10:9 0 ------
5:4 0 ------
25:24 0 dicot
27:25 0.210309918 beep
128:125 0.227535398 augmented
32805:32768 0.3472592 schismic
250:243 0.35424875 porcupine
135:128 0.396697481 pelogic
2048:2025 0.41184295 diaschismic
15625:15552 0.444302056 kleismic
256:243 0.466943504 blackwood
648:625 0.487048023 diminished
32:27 0.488324507 ------
3125:3072 0.493376213 magic

Those with no name have such high error that they can't reasonably be
considered 5-limit temperaments. The same is perhaps true of 'beep'
and 'father'. The other temperaments (including dicot, arguably error
too high, which conflates major and minor thirds) have all been used
in music -- meantone by most Western musicians in the Renaissance,
Baroque, and early & middle Classical periods; augmented and
diminished by 12-equal composers since the end of the Romantic
period; porcupine by Herman Miller; kleismic by Mats Oljare;
blackwood by Blackwood; magic by Graham Breed, and I've used pelogic
and diaschismic.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> The
> definitions given on the limits page require neither p to be prime

Look again. As of recently, the prime-limit definition *does* require
n (or p, if we prefer) to be prime. The odd-limit definition

> nor o to
> be odd

Again, look again (and ignore Monz's word-order inversion).

on 12/2/03 1:41 AM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> The
>> definitions given on the limits page require neither p to be prime
>
> Look again. As of recently, the prime-limit definition *does* require
> n (or p, if we prefer) to be prime. The odd-limit definition
>
>> nor o to
>> be odd
>
> Again, look again (and ignore Monz's word-order inversion).

I see. I must have had an un-refreshed web page sitting around.

This still does not change my point about wanting the page to be have the
form of a definition for <n>-limit. Did you not get my point there? Am I
being too formal?

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 12/2/03 1:41 AM, Paul Erlich <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> >> The
> >> definitions given on the limits page require neither p to be
prime
> >
> > Look again. As of recently, the prime-limit definition *does*
require
> > n (or p, if we prefer) to be prime. The odd-limit definition
> >
> >> nor o to
> >> be odd
> >
> > Again, look again (and ignore Monz's word-order inversion).
>
> I see. I must have had an un-refreshed web page sitting around.
>
> This still does not change my point about wanting the page to be
have the
> form of a definition for <n>-limit. Did you not get my point
there? Am I
> being too formal?
>
> -Kurt

Ask yourself, "If I read something that says 'blah 5-limit blah blah
blah 11-limit blah 15-limit . . .', and I reach for a dictionary, I
look under . . ."

blah?

on 12/2/03 2:17 AM, Paul Erlich <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 12/2/03 1:41 AM, Paul Erlich <paul@s...> wrote:
>>
>>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>>>
>>>> The
>>>> definitions given on the limits page require neither p to be
> prime
>>>
>>> Look again. As of recently, the prime-limit definition *does*
> require
>>> n (or p, if we prefer) to be prime. The odd-limit definition
>>>
>>>> nor o to
>>>> be odd
>>>
>>> Again, look again (and ignore Monz's word-order inversion).
>>
>> I see. I must have had an un-refreshed web page sitting around.
>>
>> This still does not change my point about wanting the page to be
> have the
>> form of a definition for <n>-limit. Did you not get my point
> there? Am I
>> being too formal?
>>
>> -Kurt
>
> Ask yourself, "If I read something that says 'blah 5-limit blah blah
> blah 11-limit blah 15-limit . . .', and I reach for a dictionary, I
> look under . . ."
>
> blah?

Limit of course. Which is the reason the page must be listed under L. But
if you must push me to my formal best I would advocate the form of an
implicit context-sensitive definition, e.g.

limit

<n>-limit

1 blah1 <n>-limit blah2

2 blah3 <n>-limit blah4

In other words limit is being defined implicitly based on its involvement in
the form <n>-limit. It actually can not be defined otherwise. Or rather,
other definitions of "limit" would be a separate point. For example one
might discuss other notions of "limit" that do not depend on usage in the
form <n>-limit.

As a formalist, I would want to look up <n>-limit in the first place, just
as Mathematica would do in pattern matching, for example. But
english-language dictionaries to not support such formalisms. So <n>-limit
would have to be found under "limit". Indeed, Mathematica does the very
same thing, associating all patterns involving a given symbol with the
symbol itself. Yet the pattern definitions retain their forms and do not
try to be definitions of the symbol without the form.

I also advocate that adherence to formalisms should be done only to the
extent that it does not interfere significantly with ordinary interpretation
by ordinary people. For those purposes I also like to see redundant cues,
and reshaping of formalisms so that the result can serve both the formalists
and everyone else (and a given person can fall in both categories at the
same time) to maximal advantage.

-Kurt

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> but it's late and i'm too tired to do it now ... if you'd
> like to make an addition to the definitions i'll be happy
> to add them in tomorrow! ;-)

I made a huge post on this but for some reason it hasn't appeared on
the list yet . . . maybe I sent it to you personally? I don't
remember . . .

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > right ... this is another reason why i wanted to leave it
> > > as "n-limit" instead of changing it to the less-generic
> > > "p-limit" and "o-limit".
> >
> > Why are those less generic??
> >
> > > well, you were certainly right about that. thanks to
> > > your post, i saw that i had used the italic-n in definition 1
> > > (prime-limit) but used a plain-n in definition 2 (odd-limit).
> > > i've made it consistently italic now. thanks.
> >
> > Monz, I think *both* definitions need to be modified to
> > allow for the fact that we're often referring to these
> > ratios' approximations, rather than strictly their
> > just-intonation ideals, when we use the term 'limit'.
>
>
> yep, that sounds right ... and i saw your other recent
> posts on this subject.
>
> but it's late and i'm too tired to do it now ... if you'd
> like to make an addition to the definitions i'll be happy
> to add them in tomorrow! ;-)

paul is the *man*!!!!

thanks to him, we now have a wonderful and very detailed
definition of "limit", with extensive lists of examples of
each case of the use of the word "limit":

http://sonic-arts.org/dict/limit.htm

bravo!

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> paul is the *man*!!!!
>
> thanks to him, we now have a wonderful and very detailed
> definition of "limit", with extensive lists of examples of
> each case of the use of the word "limit":

whew, i thought it might have disappeared.

> http://sonic-arts.org/dict/limit.htm
>
> bravo!

Thanks monz. The word "conisistent" appears near the end; it should
be spelled "consistent".

Now, can you change the definition of "saturated" so that
the "intervallic" sense of limit is explicitly specified? Change "odd
limit" to "intervallic odd limit" in the first sentence of
<http://www.sonic-arts.org/dict/saturat.htm>.

Thanks!

hi paul,

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > paul is the *man*!!!!
> >
> > thanks to him, we now have a wonderful and very detailed
> > definition of "limit", with extensive lists of examples of
> > each case of the use of the word "limit":
>
> whew, i thought it might have disappeared.

you did send it to me privately. i just got it this morning.

> > http://sonic-arts.org/dict/limit.htm
> >
> > bravo!
>
> Thanks monz. The word "conisistent" appears near the end; it should
> be spelled "consistent".
>
> Now, can you change the definition of "saturated" so that
> the "intervallic" sense of limit is explicitly specified?
> Change "odd limit" to "intervallic odd limit" in the
> first sentence of
> <http://www.sonic-arts.org/dict/saturat.htm>.
>
> Thanks!

done. thank *you*!

-monz

Heya Paul,

>If we want a finite list, Gene tells us that Baker's Theorem (I wish
>I knew more) assures us that we can use a 'badness' function
>called 'epimericity', which is
>
>log(|n-d|)/log(d), or just as well, log(|n-d|)/log(n)
>
>This can be expressed as
>
>log("avg. error")/"complexity" + log("complexity")/"complexity" + 1

Here you lost me. Isn't

log(|n-d|)/log(d)

just

log("avg. error")/"complexity"

? Where's the

+ log("complexity")/"complexity" + 1

bit coming from?

-Carl

>I just spent a couple of hours scratching my head -- this appears at
>first to be a *decreasing* function of complexity -- but it isn't
>since log("avg. error") will almost always be negative!
>
>Anyway, if we toss out epimericities above 0.5, and 'commas' larger
>than 600 cents, here's the list of 5-limit commas, epimericities, and
>names of temperaments (from the database above)
>
>
>16:15 0 father
>6:5 0 ------
>81:80 0 meantone
>4:3 0 ------
>9:8 0 ------
>10:9 0 ------
>5:4 0 ------
>25:24 0 dicot
>27:25 0.210309918 beep
>128:125 0.227535398 augmented
>32805:32768 0.3472592 schismic
>250:243 0.35424875 porcupine
>135:128 0.396697481 pelogic
>2048:2025 0.41184295 diaschismic
>15625:15552 0.444302056 kleismic
>256:243 0.466943504 blackwood
>648:625 0.487048023 diminished
>32:27 0.488324507 ------
>3125:3072 0.493376213 magic

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Heya Paul,
>
> >If we want a finite list, Gene tells us that Baker's Theorem (I
wish
> >I knew more) assures us that we can use a 'badness' function
> >called 'epimericity', which is
> >
> >log(|n-d|)/log(d), or just as well, log(|n-d|)/log(n)
> >
> >This can be expressed as
> >
> >log("avg. error")/"complexity" + log("complexity")/"complexity" + 1
>
> Here you lost me. Isn't
>
> log(|n-d|)/log(d)
>
> just
>
> log("avg. error")/"complexity"

no, since "avg. error", otherwise known as "heuristic error", is

|n-d|/(d*log(d)),

and "complexity", otherwise known as "heuristic complexity", is

log(d),

then your expression

log("avg. error")/"complexity"

would be

log(|n-d|/(d*log(d)))/log(d)

or

(log(|n-d|) - log(d) - log(log(d)))/log(d)

or

log(|n-d|)/log(d) - 1 - log(log(d))/log(d)

or

epimericity - 1 - log("complexity")/"complexity"

which is not what we want.

> ? Where's the
>
> + log("complexity")/"complexity" + 1
>
> bit coming from?

By now you should see it -- I've taken you within a step of the final
result, since I've isolated the epimericity term above. Still don't
see it? Tuning-math, anyone?

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> Hey Gene, I think I've demonstrated a case where it's actually
useful
> to say 'odd-limit' and 'prime-limit'!

But you didn't find it useful to say "odd-limit interval".

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48658.html#48872

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> >
> > > Which brings to mind a question for Gene: Are there any ETs
where
> > > the 'best' val uses different intervals for 3:1 and 9:3,
> > where 'best'
> > > is defined by *my* odd-limit rms error calculation?
> >
> > I would expect so; we could certainly look.
>
> I think that's a good idea -- we shouldn't be so "prime-focused"
when
> defining vals or anything else.

***Looking over Monz' page again, it seems pretty evident that the
*odd* limit is what I have been familiar with, in the Doty primer, in
Partch, and in discussions on this list.

So, I'm not quite so certain as the importance of the *prime* limits..

??

J. Pehrson

>***Looking over Monz' page again, it seems pretty evident that the
>*odd* limit is what I have been familiar with, in the Doty primer, in
>Partch, and in discussions on this list.

Actually IIRC Doty's primer uses prime limits.

-Carl

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> Thanks monz. The word "conisistent" appears near the end; it should
> be spelled "consistent".

Also, "prime number n" should be changed to "prime number p". Prime
numbers are not called "n"!

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > Thanks monz. The word "conisistent" appears near the end;
> > it should be spelled "consistent".
>
> Also, "prime number n" should be changed to "prime number p".
> Prime numbers are not called "n"!

in *my* Dictionary they are! ;-)

i guess you missed the whole exchange we had about this:

/tuning/topicId_48658.html#48917

i deliberately avoided calling it "prime number p" because
in definition #2 i would have had to use "odd number o" for
consistency, and i didn't like the look of that.

the whole point is that theorists frequently use "n-limit"
(where "n" is usually an actual number and not a variable)
without the qualifier (contrary to my recommendation),
and having to keep "p-limit" and "o-limit" separate would
just lead to more confusion.

try to get used to the idea of "prime number n", unless
you have a better suggestion.

-monz

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_48658.html#48872
>
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > > --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > >
> > > > Which brings to mind a question for Gene: Are there any ETs
> where
> > > > the 'best' val uses different intervals for 3:1 and 9:3,
> > > where 'best'
> > > > is defined by *my* odd-limit rms error calculation?
> > >
> > > I would expect so; we could certainly look.
> >
> > I think that's a good idea -- we shouldn't be so "prime-focused"
> when
> > defining vals or anything else.
>
>
> ***Looking over Monz' page again, it seems pretty evident that the
> *odd* limit is what I have been familiar with, in the Doty primer,
in
> Partch, and in discussions on this list.

No, Doty only uses prime limit, as I recall.

on 12/3/03 5:04 AM, monz <monz@attglobal.net> wrote:

> hi Gene,
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
>> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>>
>>> Thanks monz. The word "conisistent" appears near the end;
>>> it should be spelled "consistent".
>>
>> Also, "prime number n" should be changed to "prime number p".
>> Prime numbers are not called "n"!
>
>
>
> in *my* Dictionary they are! ;-)
>
> i guess you missed the whole exchange we had about this:
>
> /tuning/topicId_48658.html#48917
>
>
> i deliberately avoided calling it "prime number p" because
> in definition #2 i would have had to use "odd number o" for
> consistency, and i didn't like the look of that.
>
> the whole point is that theorists frequently use "n-limit"
> (where "n" is usually an actual number and not a variable)
> without the qualifier (contrary to my recommendation),
> and having to keep "p-limit" and "o-limit" separate would
> just lead to more confusion.

I have to say in a world where everyone thinks differently it is a relief to
*occasionally* find someone who agrees, and even gives a similar explanation
for the choice. Of course too much agreement would be boring.

-Kurt

>
> try to get used to the idea of "prime number n", unless
> you have a better suggestion.
>
>
>
> -monz

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48658.html#48992

> >
> > ***Looking over Monz' page again, it seems pretty evident that
the
> > *odd* limit is what I have been familiar with, in the Doty
primer,
> in
> > Partch, and in discussions on this list.
>
> No, Doty only uses prime limit, as I recall.

***Hmmm. I can't believe I'm still not understanding this, but I'm
past embarassment, so I'll plow forward... :)

So, the *ODD* limit looks like it's the more *inclusive* one, or,
rather, it contains more sonorities because it uses all the odd
numbers rather than just "primes..." ??

And the *PRIME* limit is the restrictive one that focuses on the
basic intervals of just like the perfect fifth for the 3-limit, the
major third for the 5-limit and the minor seventh for the 7-limit.
So, it's used in a more *general* sense to explain "types" of just
intonation ??

help!

Tx,

Joseph

>***Hmmm. I can't believe I'm still not understanding this, but I'm
>past embarassment, so I'll plow forward... :)
>
>So, the *ODD* limit looks like it's the more *inclusive* one, or,
>rather, it contains more sonorities because it uses all the odd
>numbers rather than just "primes..." ??
>
>And the *PRIME* limit is the restrictive one that focuses on the
>basic intervals of just like the perfect fifth for the 3-limit, the
>major third for the 5-limit and the minor seventh for the 7-limit.
>So, it's used in a more *general* sense to explain "types" of just
>intonation ??
>
>help!

What's a good exercise JP, is trying to list for yourself all
the 5-odd-limit, and then all the 5-prime-limit ratios. Refer
to monz's dictionary for help, since it represents at this point
the best explanation we know how to give (and was updated
recently).

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> What's a good exercise JP, is trying to list for yourself all
> the 5-odd-limit, and then all the 5-prime-limit ratios. Refer
> to monz's dictionary for help, since it represents at this point
> the best explanation we know how to give (and was updated
> recently).

This is a really, really bad idea, since there are an infinite number
of 5-prime-limit ratios. The odd limit is about consonances, the
prime limit is not.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

/tuning/topicId_48658.html#49246

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > What's a good exercise JP, is trying to list for yourself all
> > the 5-odd-limit, and then all the 5-prime-limit ratios. Refer
> > to monz's dictionary for help, since it represents at this point
> > the best explanation we know how to give (and was updated
> > recently).
>
> This is a really, really bad idea, since there are an infinite
number
> of 5-prime-limit ratios.

***Hmmm... I gather this exercise is going to take me quite some time
then....

JP

>> What's a good exercise JP, is trying to list for yourself all
>> the 5-odd-limit, and then all the 5-prime-limit ratios. Refer
>> to monz's dictionary for help, since it represents at this point
>> the best explanation we know how to give (and was updated
>> recently).
>
>This is a really, really bad idea, since there are an infinite number
>of 5-prime-limit ratios.

Great Gene, thanks for ruining my exercise.

-Carl

>>This is a really, really bad idea, since there are an infinite
>>number of 5-prime-limit ratios.

Also of 5-odd-limit ratios, though most of them are not in
a musical register.

> Great Gene, thanks for ruining my exercise.

Prob. should have had a :) on that. :)

-Carl

on 12/7/03 6:50 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> What's a good exercise JP, is trying to list for yourself all
>>> the 5-odd-limit, and then all the 5-prime-limit ratios. Refer
>>> to monz's dictionary for help, since it represents at this point
>>> the best explanation we know how to give (and was updated
>>> recently).
>>
>> This is a really, really bad idea, since there are an infinite number
>> of 5-prime-limit ratios.
>
> Great Gene, thanks for ruining my exercise.

I was thinking the same thing - "he will find that out for himself".

> -Carl

on 12/7/03 7:05 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> This is a really, really bad idea, since there are an infinite
>>> number of 5-prime-limit ratios.
>
> Also of 5-odd-limit ratios, though most of them are not in
> a musical register.
>
>> Great Gene, thanks for ruining my exercise.
>
> Prob. should have had a :) on that. :)

Me too. :)

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_48658.html#48992
>
> > >
> > > ***Looking over Monz' page again, it seems pretty evident that
> the
> > > *odd* limit is what I have been familiar with, in the Doty
> primer,
> > in
> > > Partch, and in discussions on this list.
> >
> > No, Doty only uses prime limit, as I recall.
>
>
> ***Hmmm. I can't believe I'm still not understanding this, but I'm
> past embarassment, so I'll plow forward... :)
>
> So, the *ODD* limit looks like it's the more *inclusive* one, or,
> rather, it contains more sonorities because it uses all the odd
> numbers rather than just "primes..." ??

No, it's quite exclusive, giving *only* the ratios listed in the
Tonality Diamond diagram for each odd limit (Partch's book has the 5-
limit, 11-limit, and I believe 13-limit Tonality Diamonds -- take a
look -- you can easily construct the 7-limit and 9-limit ones now, on
your own . . .)

> And the *PRIME* limit is the restrictive one that focuses on the
> basic intervals of just like the perfect fifth for the 3-limit, the
> major third for the 5-limit and the minor seventh for the 7-limit.

It's less restrictive (even 3-limit, but also all higher prime
limits, contain an infinite number of ratios between any two ratios
you care to name, no matter how close), since all conceivable
combinations of multiplying and dividing 1/1 by these 'basic'
intervals, using each as many times as you wish, still yield ratios
belonging to the same prime limit -- in this case, the largest prime
factor of all the ratios is still 7. You'll see Doty uses this
definition of limit.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48658.html#49322

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> >
> > /tuning/topicId_48658.html#48992
> >
> > > >
> > > > ***Looking over Monz' page again, it seems pretty evident
that
> > the
> > > > *odd* limit is what I have been familiar with, in the Doty
> > primer,
> > > in
> > > > Partch, and in discussions on this list.
> > >
> > > No, Doty only uses prime limit, as I recall.
> >
> >
> > ***Hmmm. I can't believe I'm still not understanding this, but
I'm
> > past embarassment, so I'll plow forward... :)
> >
> > So, the *ODD* limit looks like it's the more *inclusive* one, or,
> > rather, it contains more sonorities because it uses all the odd
> > numbers rather than just "primes..." ??
>
> No, it's quite exclusive, giving *only* the ratios listed in the
> Tonality Diamond diagram for each odd limit (Partch's book has the
5-
> limit, 11-limit, and I believe 13-limit Tonality Diamonds -- take a
> look -- you can easily construct the 7-limit and 9-limit ones now,
on
> your own . . .)
>
> > And the *PRIME* limit is the restrictive one that focuses on the
> > basic intervals of just like the perfect fifth for the 3-limit,
the
> > major third for the 5-limit and the minor seventh for the 7-limit.
>
> It's less restrictive (even 3-limit, but also all higher prime
> limits, contain an infinite number of ratios between any two ratios
> you care to name, no matter how close), since all conceivable
> combinations of multiplying and dividing 1/1 by these 'basic'
> intervals, using each as many times as you wish, still yield ratios
> belonging to the same prime limit -- in this case, the largest
prime
> factor of all the ratios is still 7. You'll see Doty uses this
> definition of limit.

***Thanks, Paul. This helps!

JP