64:75:96

>
> From: "David J. Finnamore" <daeron@bellsouth.net>
>
> Monz wrote:
>
> > So, knowing that 64:75 was pretty close in size to the
> > discarded 6:7, I gave it a shot, and bingo!, it gave
> > exactly the sound I was looking for: the 'dark' harmonic
> > quality without the melodic 'pain'. This triad gives a
> > 'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
> > of 25:32 = ~427 cents.
>
> You tempered it. One could hardly ask for a clearer example of the
> concept of tempering. The fact that it's rational is incidental.
>

Not so fast there! He said "melodic 'pain'". If the sequence
leading to '75' was such that all the melodic movement was
required to be 5-limit to have the proper relationship with
near-term memory (or reverberation) then '75' is the 'right'
answer.

And its pretty significantly far away from 7:6, 6:5 and 19:16.
The closest lower pockets I could find to fall in were 27:23
and 34:29 which, though probably indistinguishable as dyads
from 75:64, wouldn't seem to fit into any other composerly
conception at that moment if the piece is potherwise 5-limit.

No, I won't answer the 'what is JI question'.

Bob Valentine

--- In tuning@egroups.com, Robert C Valentine <BVAL@I...> wrote:
> >
> > From: "David J. Finnamore" <daeron@b...>
> >
> > Monz wrote:
> >
> > > So, knowing that 64:75 was pretty close in size to the
> > > discarded 6:7, I gave it a shot, and bingo!, it gave
> > > exactly the sound I was looking for: the 'dark' harmonic
> > > quality without the melodic 'pain'. This triad gives a
> > > 'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
> > > of 25:32 = ~427 cents.
> >
> > You tempered it. One could hardly ask for a clearer example of
the
> > concept of tempering. The fact that it's rational is incidental.
> >
>
> Not so fast there! He said "melodic 'pain'". If the sequence
> leading to '75' was such that all the melodic movement was
> required to be 5-limit to have the proper relationship with
> near-term memory (or reverberation) then '75' is the 'right'
> answer.

Plausible, I'll admit. But he also said that 75:64 was not the only
right answer as far as harmonic quality was concerned. More to the
point, he said that 7:6 was the third that provided ideal chord
color. No factor of 5 in 7 or 6. So the need to be 5-limit is
clearly not the reason for using 75:64 here. I guess I should have
included more of his quote but I like to keep bandwidth down where
feasible.

The only reason he pulled it North was to keep the melodic intervals
smoother. That's the primary (possibly only) motivation to use
tempering in this kind of situation. (The other reasons for using
temperaments don't apply to compositions which seek use the purest
ratios possible.) Any minor third narrow enough to keep the chord
suitibly dark, and wide enough to keep the pertinent melodic line
smooth, would have done the trick whether or not it could be analyzed
as a ratio with odd limit <105 - or wherever you choose to draw the
line.

> And its pretty significantly far away from 7:6, 6:5 and 19:16.
> The closest lower pockets I could find to fall in were 27:23
> and 34:29 which, though probably indistinguishable as dyads
> from 75:64, wouldn't seem to fit into any other composerly
> conception at that moment if the piece is potherwise 5-limit.

I seem to be taking the opposite side of my recent argument, here
8-) But 27:23 and 34:29, with odd limits of 621 and 493 respectively
(unlike the offered example of 13:10), are a long way from anything
anyone has yet shown to be recognizable as pockets. As you said
yourself, they're probably indistinguishable from 75:64 in most
situations, except by the extended reference that Joe mentioned. If
he shows how the context provides that extended reference, then we're
getting somewhere. But he states that he chose it using methods that
ammount to temperament (and "composerly conception"), not methods of
rational context.

If you stack up a pair of 5:4s (1/1 5/4 25/16), then add a 3:2 on top
(resulting in 75/64), then contract the two inner tones to a unison
3/2 while sustaining the 75/64, then maybe you've got extended
reference, depending on where you came from and where you go from
there. But simply placing a 75:64 in a triad in a 5- or 7-limit
context doesn't necessarilly make it a just interval. (Or, if you
prefer it as a verb, it doesn't mean you "justed" the interval.)

David Finnamore

David Finnamore,

In http://www.egroups.com/message/tuning/15991

you need to replace every ocurrence of "odd-limit" with "complexity".
And the verb you wanted is "justly intoned".

Complexity(a:b) = a*b
Odd_limit(a:b) = Max(Remove_factors_of_2(a), Remove_factors_of_2(b))
Prime_limit(a:b) = Greatest_prime_factor(a*b)

Prime_limit is completely useless as a measure of justness (or rather
unjustness) of a ratio since it takes no account of what power the
primes might be raised to. Odd_limit assumes total octave equivalence
so it isn't really valid either, but it is often a useful
simplification.

Otherwise I agree with what you said. To establish "extended
reference" Monz needs to show how the middle note of the 64:75:96 is
related to some local tonic by a ratio simple enough to be perceived
as Just when the notes do not sound together.

Regards,
-- Dave Keenan

--- In tuning@egroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> David Finnamore,
>
> In http://www.egroups.com/message/tuning/15991
>
> you need to replace every ocurrence of "odd-limit" with
"complexity".
>
> Complexity(a:b) = a*b
> Odd_limit(a:b) = Max(Remove_factors_of_2(a), Remove_factors_of_2(b))
> Prime_limit(a:b) = Greatest_prime_factor(a*b)

OK, I did use the term Odd limit per this definition, so I assume
that what you mean is that "complexity" is a more appropriate term
than "odd limit" in the context. Then, "...would have done the trick
whether or not it could be analyzed as a ratio that falls within
whatever complexity limit you choose" would be better? And "...27:23
and 34:29, with complexities of 621 and 986 respectively (unlike the
offered example of 13:10), are a long way from anything anyone has
yet shown to be recognizable as pockets" would be better? Since
we're talking about a single triad here, why not assume octave
equivalency? Would voicing it over an octave and a quarter make that
much difference how one tempered it, or how easily it could be tuned
by ear?

> And the verb you wanted is "justly intoned".

Yes, of course - adverb and verb, actually. That was a good-natured
jab in the ribs for Carl L. and Robert W., who say that "just" is a
verb, without offering any counter-examples of its usage as anything
other than an adjective.

> Prime_limit is completely useless as a measure of justness

Sounds reasonable. Prime limit is probably useless for saying
anything about individual, isolated intervals. It can say something
about the tone color of a composition in a tuning system based on a
chain of a simple interval, as in Margo's usage of terms like
"threeness" and "sevenness." Beyond that, I don't know of a use for
it.

> Otherwise I agree with what you said. To establish "extended
> reference" Monz needs to show how the middle note of the 64:75:96
is
> related to some local tonic by a ratio simple enough to be
perceived
> as Just when the notes do not sound together.

Local tonic. That should do it.

Monz, sorry about the bluntness of my original statement about this.
I'll have to plead temporary sleepiness. I should have at least said
that it was in no way a criticism. I think your tactic was in good
taste. Tempering is, to me, often preferable to rigid just
intonation in tonal music. I'm in the habit of making a lattice for
each piece of music to see how many fifths I can keep pure, if any,
then I figure out which thirds can be pure, if any, and so forth,
then temper only what needs it, and only as far as necessary. I
prefer that to sticking it in Pythagorean, a standard meantone, or a
Well, and letting the chips fall where they may, even though it takes
a lot more time. The pure fifths, especially, are worth the effort
to me (1200 EDO fifths, actually).

David Finnamore

David Finnamore wrote:

> > Complexity(a:b) = a*b
> > Odd_limit(a:b) = Max(Remove_factors_of_2(a),
Remove_factors_of_2(b))
> > Prime_limit(a:b) = Greatest_prime_factor(a*b)
>
> OK, I did use the term Odd limit per this definition, so I assume
> that what you mean is that "complexity" is a more appropriate term
> than "odd limit" in the context.

Er. No. It's the _only_ appropriate term. You'd better look at the
definition of odd-limit again. Or look it up under "limit" in Monzo's
dictionary (which everyone on this list should have bookmarked).
http://www.ixpres.com/interval/dict/index.htm
My apologies to Robert Walker and the list for failing to look up
octony in it yesterday.

The odd-limit of 27:23 is 27, not 621. The odd-limit of 34:29 is 29,
not 986.

> > And the verb you wanted is "justly intoned".
>
> Yes, of course - adverb and verb, actually.

Ok. Verb phrase. You're at least as much of a pedant as I am. :-)

> That was a good-natured
> jab in the ribs for Carl L. and Robert W., who say that "just" is a
> verb, without offering any counter-examples of its usage as anything
> other than an adjective.

I thought that might be the case. They might need to brush up on their
grammar, but we know what they meant.

Regards,
-- Dave Keenan

--- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:

> http://www.egroups.com/message/tuning/15991
>
> --- In tuning@egroups.com, Robert C Valentine <BVAL@I...> wrote:
> > >
> > > From: "David J. Finnamore" <daeron@b...>
> > >
> > > Monz wrote [regarding the piece _3 Plus 4_]:
> > >
> > > > So, knowing that 64:75 was pretty close in size to the
> > > > discarded 6:7, I gave it a shot, and bingo!, it gave
> > > > exactly the sound I was looking for: the 'dark' harmonic
> > > > quality without the melodic 'pain'. This triad gives a
> > > > 'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
> > > > of 25:32 = ~427 cents.
> > >
> > > You tempered it. One could hardly ask for a clearer example
> > > of the concept of tempering. The fact that it's rational is
> > > incidental.
> > >
> >
> > Not so fast there! He said "melodic 'pain'". If the sequence
> > leading to '75' was such that all the melodic movement was
> > required to be 5-limit to have the proper relationship with
> > near-term memory (or reverberation) then '75' is the 'right'
> > answer.
>
> Plausible, I'll admit. But he also said that 75:64 was not
> the only right answer as far as harmonic quality was concerned.
> More to the point, he said that 7:6 was the third that provided
> ideal chord color. No factor of 5 in 7 or 6. So the need to
> be 5-limit is clearly not the reason for using 75:64 here. I
> guess I should have included more of his quote but I like to
> keep bandwidth down where feasible.
>
> The only reason he pulled it North was to keep the melodic
> intervals smoother. That's the primary (possibly only)
> motivation to use tempering in this kind of situation. (The
> other reasons for using temperaments don't apply to compositions
> which seek use the purest ratios possible.) Any minor third
> narrow enough to keep the chord suitibly dark, and wide enough
> to keep the pertinent melodic line smooth, would have done the
> trick whether or not it could be analyzed as a ratio with odd
> limit <105 - or wherever you choose to draw the line.

I see that several others have already responded to this, but
I'm replying here before reading their comments.

David F., I've already agreed with you somewhat that, yes, in a
sense I was tempering the 7:6 here. *But*, IMO... I specifically
chose 75:64 instead of 7:6 *not only* because of the desire
to reduce *melodic* 'pain', but also because I think I wanted
to reduce 'prime-limit pain'.

Paul Erlich and I have had a lot of disagreements about the
importance of primes in tuning, but after *much* debate, we
have finally come to agreement that the real importance of
primes lies in the fact that in a general, system sense (i.e.,
not considering specific dyads), each new prime (up to some rather
low but not clearly-defined limit, probably somewhere between
13 and 23) introduces a new 'affect' into the sound of a tuning
which creates a new sonic dimension.

In this particular case, I really believe that most of the
'pain' I was hearing from the 7:6 had to do with the fact that
the factor 7 was introducing a new dimension into the sound
that conflicted with the ratios I had already used up to that
point. The two reasons I came to this conclusion are: 1) the
5-limit 75:64 is so close to 7:6 but yet sounds so different, and
2) I have a previous experiment where I retuned the beginning
of Satie's _Sarabande No. 1_ (with lots of unresolved 'major 7th'
and 'dominant 9th' chords) in many different ways, and the
versions that use 7-limit ratios clearly sound 'out of tune'
in many respects.

Thus, I would say that both you *and* Bob are correctly
characterizing my reasons for my choice of tuning here.

> ...But 27:23 and 34:29, with odd limits of 621 and 493
> respectively (unlike the offered example of 13:10), are a
> long way from anything anyone has yet shown to be recognizable
> as pockets. As you said yourself, they're probably
> indistinguishable from 75:64 in most situations, except
> by the extended reference that Joe mentioned. If he shows
> how the context provides that extended reference, then we're
> getting somewhere. But he states that he chose it using
> methods that ammount to temperament (and "composerly
> conception"), not methods of rational context.
>
> If you stack up a pair of 5:4s (1/1 5/4 25/16), then add a
> 3:2 on top (resulting in 75/64), then contract the two inner
> tones to a unison 3/2 while sustaining the 75/64, then maybe
> you've got extended reference, depending on where you came
> from and where you go from there. But simply placing a 75:64
> in a triad in a 5- or 7-limit context doesn't necessarilly make
> it a just interval. (Or, if you prefer it as a verb, it doesn't
> mean you "justed" the interval.)

David F., your 'extended reference' analysis of this is basically
correct... I tend to think of this particular 75:64 as the
5:4 above the 15:16 'leading-tone' of the chord-root, which
is just another way of saying the same thing you said here...
a different route along the 5-limit map, if you will.

I think it's very possible that I was thinking along the lines
of 'chord substitution' as described by Schoenberg in his
_Harmonielehre_: substituting certain notes of a particular
chord with notes which 'belong' to a different chord which
bears a clear relationship to that first chord. In this case,
instead of using the usual 'minor 3rd' for a tonic (i) minor
chord (which in standard JI theory is a 6/5), I'm using the
'major 3rd' of the VII chord as a substitute.

Note also that I was using some compositional sleight-of-hand
here by 'tonicizing' the D#-minor chord momentarily; in the
overall context of the tune, D#-minor is really the vi
(submediant) of the overall tonic F#-major.

Something else that should be considered in discussions of this
by other Listers is that in my own pieces and retunings of music
by others, I *never* consider odd-limit; I always think strictly
in terms of prime-limit.

Honestly, to say anything more detailed about my compositional
choice here would require me to go back and study the lattice
and the MIDI-file in detail, in order to consider the ratios I
used in the surrounding chords and melodic lines. This is
something I simply can't do right now...

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

> > Complexity(a:b) = a*b

Of course, there are many different definitions of complexity:

harmonic distance, indigestibility, harmonic complexity, harmonic entropy,
harmonicity, gradus suavitatis

(http://www.ixpres.com/interval/dict/complex.htm)

Of these only harmonic distance (Tenney Harmonic Distance) is a function of
a*b -- it's actually log(a*b) where the base of the log is 2. But I think
Dave K. is getting at the idea that there are three factors which determine
the consonance/dissonance of an interval,

complexity
tolerance
span

and I think I've convinced Dave K. that, in this context, Tenney's
complexity, or a monotonic function of it such as a*b, is the best measure
of complexity. And I was led there by Graham Breed. So perhaps we should
call a*b the Tenney/Breed/Erlich/Keenan complexity or something, just to
avoid confusing with Wilson's harmonic complexity or other functions.

Monz wrote,

>David F., I've already agreed with you somewhat that, yes, in a
>sense I was tempering the 7:6 here. *But*, IMO... I specifically
>chose 75:64 instead of 7:6 *not only* because of the desire
>to reduce *melodic* 'pain', but also because I think I wanted
>to reduce 'prime-limit pain'.

As you know, I think prime-limit pain is a figment of your (and many other
fine minds') imagination.

>Paul Erlich and I have had a lot of disagreements about the
>importance of primes in tuning, but after *much* debate, we
>have finally come to agreement that the real importance of
>primes lies in the fact that in a general, system sense (i.e.,
>not considering specific dyads), each new prime (up to some rather
>low but not clearly-defined limit, probably somewhere between
>13 and 23) introduces a new 'affect' into the sound of a tuning
>which creates a new sonic dimension.

No, I would not agree with this. What I would agree with is the purely
mathematical fact that each new prime introduces a new dimension of pitch
possibilities into an infinite JI system.

>In this particular case, I really believe that most of the
>'pain' I was hearing from the 7:6 had to do with the fact that
>the factor 7 was introducing a new dimension into the sound
>that conflicted with the ratios I had already used up to that
>point.

I think you're hallucinating, or you've brainwashed yourself with
prime-obsession (no disrespect intended).

>The two reasons I came to this conclusion are: 1) the
>5-limit 75:64 is so close to 7:6 but yet sounds so different,

We've already explained that in previous messages. Anyway, it's irrelevant
-- one can come up with any number of ratios with _any_ desired prime limit
(5, 7, 23, whatever) that will be as close to 75:64 as to be audibly
indistinguishable.

>and
>2) I have a previous experiment where I retuned the beginning
>of Satie's _Sarabande No. 1_ (with lots of unresolved 'major 7th'
>and 'dominant 9th' chords) in many different ways, and the
>versions that use 7-limit ratios clearly sound 'out of tune'
>in many respects.

What does that have to do with this???

>David F., your 'extended reference' analysis of this is basically
>correct... I tend to think of this particular 75:64 as the
>5:4 above the 15:16 'leading-tone' of the chord-root, which
>is just another way of saying the same thing you said here...
>a different route along the 5-limit map, if you will.

Well that's another matter altogether . . . so what you're saying is that
the sonorousness of the 64:75:96 chord was not a relevant factor to you
here, what you wanted was a particular _melodic_ relationship . . . now as
you know I don't thing 5:4 is any kind of _melodic_ attractor, BUT, if
you're using a lot of melodic 5:4s in your piece, then it would make sense
MOTIVICALLY to use the 75:64 IF it was forming a melodic 5:4 with the 15/16
. . . is it?

>Honestly, to say anything more detailed about my compositional
>choice here would require me to go back and study the lattice
>and the MIDI-file in detail, in order to consider the ratios I
>used in the surrounding chords and melodic lines. This is
>something I simply can't do right now...

Oh . . . oh well . . .

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/16024
>
> Monz wrote,
>
> > ...
> > Paul Erlich and I have had a lot of disagreements about the
> > importance of primes in tuning, but after *much* debate, we
> > have finally come to agreement that the real importance of
> > primes lies in the fact that in a general, system sense (i.e.,
> > not considering specific dyads), each new prime (up to some
> > rather low but not clearly-defined limit, probably somewhere
> > between 13 and 23) introduces a new 'affect' into the sound
> > of a tuning which creates a new sonic dimension.
>
> No, I would not agree with this. What I would agree with is
> the purely mathematical fact that each new prime introduces
> a new dimension of pitch possibilities into an infinite JI
> system.
>
> > In this particular case, I really believe that most of the
> > 'pain' I was hearing from the 7:6 had to do with the fact that
> > the factor 7 was introducing a new dimension into the sound
> > that conflicted with the ratios I had already used up to that
> > point.
>
> I think you're hallucinating, or you've brainwashed yourself with
> prime-obsession (no disrespect intended).
>
> > The two reasons I came to this conclusion are: 1) the
> > 5-limit 75:64 is so close to 7:6 but yet sounds so different,
>
> We've already explained that in previous messages. Anyway,
> it's irrelevant -- one can come up with any number of ratios
> with _any_ desired prime limit (5, 7, 23, whatever) that will
> be as close to 75:64 as to be audibly indistinguishable.

Paul, I think perhaps my point here was not as clear as I could
have made it; in any case, I certainly don't mean to 'put words
in your mouth'. Maybe I can clarify.

I'm simply trying to point out that considering a composition
or tuning system *as a whole*, prime-factors are important
because they introduce another dimension into the sound.
As I pointed out, there is a limit to one's perception of
this; this statement is true only of relatively low-integer
prime-factors (certainly including 2, 3, 5, 7, and 11,
and *possibly limited only to those*). And certainly, there
is also an exponent-limit involved in our perception of
this: only ratios that are quite close in lattice-space
to a given local 1/1 will be perceived as demonstrating
these multiple dimensions. As you point out, and as I
already felt I had expressed, using either higher primes
or higher exponents or both will give ratios that come
arbitrarily close to any others.

If you're in agreement with this, then what I'm saying is
simply that when listening to a piece whose tuning is full
of 5-limit ratios, a 7-limit one can 'stick out like a sore
thumb'. Again, I'm not saying that this is *always* or
*necessarily* the case, but in my experience, a sudden *emphatic*
use of 7 in a 5-limit context can sound awfully out-of-tune.
And I would say that whether the higher prime is emphasized
or not, and exactly *how* it is emphasized, has an important
bearing on its 'in/out-of-tuneness'. Any disagreement with that?

You might like to note that right after this D#-minor cadence,
the 'hook' shifts the tonic to F#-major, and I used an 11:8
in the penultimate chord which sounds weird but (to me) *not*
out-of-tune. That's the reason for my note above about *how*
the higher prime is emphasized.

>
> > and
> > 2) I have a previous experiment where I retuned the beginning
> > of Satie's _Sarabande No. 1_ (with lots of unresolved
> > 'major 7th' and 'dominant 9th' chords) in many different
> > ways, and the versions that use 7-limit ratios clearly sound
> > 'out of tune' in many respects.
>
> What does that have to do with this???

Only that it was a previous listening experience on which I drew
when selecting ratios in my retuning of _3 Plus 4_. The Satie
piece, while stylistically very different from my tune, has a
chord vocabulary that's rather similar to that which I used in
_3 Plus 4_.

>
> > David F., your 'extended reference' analysis of this is
> > basically correct... I tend to think of this particular
> > 75:64 as the 5:4 above the 15:16 'leading-tone' of the
> > chord-root, which is just another way of saying the same
> > thing you said here... a different route along the 5-limit
> > map, if you will.
>
> Well that's another matter altogether . . . so what you're
> saying is that the sonorousness of the 64:75:96 chord was
> not a relevant factor to you here, what you wanted was a
> particular _melodic_ relationship

As I said in my previous response to David F., the sonorousness
of that chord was *very much* a relevant factor, but so was
my desire to reduce 'pain', whether that pain is based on
melodic pitch-height or overall prime-ness or whatever.
My decision to go with 64:75:96 was definitely the result
of a balancing act which took several different audible
considerations into account.

> . . . now as you know I don't thing [_sic_: think] 5:4 is
> any kind of _melodic_ attractor, BUT, if you're using a lot
> of melodic 5:4s in your piece, then it would make sense
> MOTIVICALLY to use the 75:64 IF it was forming a melodic
> 5:4 with the 15/16 . . . is it?

Nope. In fact, 15/16 isn't even a part of the cadence which
leads to this 64:75:96 D#-minor chord! - I arrived at it thru
a plagal cadence with a chord-root motion of B - A#-minor -
G#-minor - D#-minor, then I even emphasize the plagal aspect
at the cadence itself with i-iv-i. The 15/16 only appears
*before* this progression as a quick passing-tone.

But I still think that the Schoenbergian kind of substitution
I mentioned is in effect here, simply because of my prolonged
tonicization of D#-minor. I think the ear/brain perceives
the relationship of 75/64 as the 'major 3rd' of the 'leading-tone'
anyway, and accepts it as a different flavor of 'minor 3rd'
because its relationship to the 1/1 (D#) is clear-cut.

Like I said and you agreed, my foreknowledge of how the 5-limit
lattice is constructed certainly affected my decsion of how
to tune this chord.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

Well Monz, I think that in terms of the sonorousness of the chord itself,
prime limit is completely irrelevant, as we've gone over many times before;
and in terms of the extended melodic reference, you've got quite a balancing
act to maneuver through to maintain the 15/8 in your head and to perceive a
dissonant chord member as the 5:4 of that long-departed 15/8, and not in
relation to a more immediate reference point instead. I really think you're
fooling yourself -- on the first point (sonorousness), I have no doubt; and
on the second point, it's hard to imagine how your big extended reference
lattice theory could take into account the fact that tempered systems were
used for, and in fact required for, most great works of Western music (for
Schoenberg, the minor third was _always_ the major third above the leading
tone since all the music he considered happened to be in 12-tET).

--- In tuning@egroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
> David Finnamore wrote:
>
> > > Complexity(a:b) = a*b
> > > Odd_limit(a:b) = Max(Remove_factors_of_2(a),
> Remove_factors_of_2(b))
> > > Prime_limit(a:b) = Greatest_prime_factor(a*b)
> >
> > OK, I did use the term Odd limit per this definition, so I assume
> > that what you mean is that "complexity" is a more appropriate
term
> > than "odd limit" in the context.
>
> Er. No. It's the _only_ appropriate term. You'd better look at the
> definition of odd-limit again. Or look it up under "limit" in
Monzo's
> dictionary (which everyone on this list should have bookmarked).
> http://www.ixpres.com/interval/dict/index.htm

Oh! So I meant _complexity_ all along. Thanks for the correction
and sorry to everyone for the misunderstaning.

David Beardsley: You were right, I was confused! :-)

David Finnamore