Odd limit

Hi George.

>Consonant interval-ratios are classified according to *odd* rather
>than *prime* limit, according to the largest *odd number* in the
>ratio. 81/64 would therefore not be considered a consonance below
>the 81-limit, so it would be a 3-limit dissonance. Likewise, 9/7 is
>a 9-limit (rather than a 7-limit) consonance.
>
>This relationship between consonance and harmonic limit, first
>proposed by Harry Partch, is backward-compatible with common-practice
>(5-limit) usage and is widely accepted in this group.
>
>--George

I understand that the fact that in all the ratios (n/d) we normally consider
n and d are coprimes can generate some confusion with primes (especially because 80% of odds in the first 20 harmonics are primes).

Anyway the odd limit is defined on the basis of a prime number which is 2. I think this comes from octave equivalence but it also gives another kind of equivalence which is that of intervals inversions: 4/3 and 3/2, 16/15 and 15/8 have the same odd limit. Why is this done only for number 2? Why we don't consider also that 6/5 is the inversion of 9/5 respect 3/2?

If we say that odd limit has to be joined with other parameters as n*d or n+d to have a good ranking for consonance we have to consider that for odd/even ratios the consonance is in inverse relation to n*d or n+d while for even/odd ratios is the opposite: try comparing 12/7 with 8/7, 12/5 with 8/5, 16/9 with 14/9 and this with 10/9.

In any case is really hard to my ears to accept that 8/7 sounds smoother than 9/4 or 16/9. This could be because prime limit works better or maybe because each odd limit has a threshold of n*d where it begins to be worst than the successive limit.

A reason why primes could be important in ranking consonance is maybe related to perception. If a ratio should be dismantled in *phisical* units, using primes we would spare room. I don't know if there is some relation between primes and perception (any papers?) but I suspect that, if intellectually we can compress data in this way, also our brain could work in the same manner.

I try to represent this (O = unit on the origin, U=other units):

x-axis: addition

_OUU_ this is 3 or 3^1 or 1+1+1.

y-axis: exponents

_UUU_
_O____ this is 9 or 3^2 or (1+1+1)^2

As you can see 4 units are enough to compress the number 9.

9/8 could be builded of 8 units:

_UUU_
_O____
_U____
_U____
_UU___

z-axis: multiplication (primes)

15 should be represented in this way:

_OUU__ (for z=0)

_UUUUU_ (for z=1)

The volume of each ratio in this tridimensional space gives the quantity of information necessary to store it.
This is the ranking for the most common ratios:

3/2 (5)
4/3 (6)
9/8 (8)
5/4 (8)
5/3 (8)
16/9 (9)
9/5 (9)
7/4 (10)
6/5 (11)
10/9 (11)
7/5 (12)
15/8 (12)
16/15 (13)

If I would be a brain I would store ratios in this way....

lorenzo

Hi Lorenzo,

> Anyway the odd limit is defined on the basis of a prime number
> which is 2. I think this comes from octave equivalence but it
> also gives another kind of equivalence which is that of
> intervals inversions: 4/3 and 3/2 ... have the same odd limit.

4/3 and 3/2 would have to be the same in any octave-equivalent
regime, wouldn't they? So this isn't another kind of equivalence,
it's a facet of octave equivalence.

> Why is this done only for number 2?

Because psychoacoustics tells us that 2 is special.

> Why we don't consider also that 6/5 is the inversion of 9/5
> respect 3/2?

The high degree of symmetry at 3/2 in the world's scales suggests
something like '3/2-equivalence' is going on. But let me ask,
how similar do 6/5 and 9/5 sound to you, compared to 3/2 and 4/3.
Better yet, can you take a number of such examples and compare
them?

> If we say that odd limit has to be joined with other parameters
> as n*d ...

Incidentally, Paul Erlich showed that if you start with n*d
and add only octave equivalence, you get odd limit.

> In any case is really hard to my ears to accept that 8/7 sounds
> smoother than 9/4 or 16/9

The "critical band" must also be considered. 8/7 is pretty
close to it, and is within it for much of the musical
pitch range.

> to have a good ranking for consonance we have to consider that
> for odd/even ratios the consonance is in inverse relation to
> n*d or n+d while for even/odd ratios is the opposite: try
> comparing 12/7 with 8/7,

I will (and get back to you).

> 12/5 with 8/5,

Interval size comes into play here.

> 16/9 with 14/9 and this with 10/9.

10/9 has even more critical band issues than 8/7.

> This could be because prime limit works better

But it doesn't. Prime limit alone leads to weird results,
since there are 3-limit commas arbitrarily close to any
interval you care to name.

> or maybe
> because each odd limit has a threshold of n*d where it begins
> to be worst than the successive limit.

If you could quantify this suggestion...

> I don't know if there is some relation
> between primes and perception (any papers?)

I suspect there is some sort of factoring going on in the
auditory system -- a hierarchy of periodicities, perhaps --
but if so it doesn't go very far out into the number system,
and is generally overwhelmed by other facets of hearing like
roughness and interval size. I seem to remember finding a
vaguely-related paper on this once. I'll try to dig it up.

> I try to represent this (O = unit on the origin, U=other units):
>
> x-axis: addition
>
> _OUU_ this is 3 or 3^1 or 1+1+1.
>
> y-axis: exponents
>
> _UUU_
> _O____ this is 9 or 3^2 or (1+1+1)^2
>
> As you can see 4 units are enough to compress the number 9.
>
> 9/8 could be builded of 8 units:
>
> _UUU_
> _O____
> _U____
> _U____
> _UU___
>
> z-axis: multiplication (primes)
>
> 15 should be represented in this way:
>
> _OUU__ (for z=0)
>
> _UUUUU_ (for z=1)
>
> The volume of each ratio in this tridimensional space gives
> the quantity of information necessary to store it.
> This is the ranking for the most common ratios:
>
> 3/2 (5)
> 4/3 (6)
> 9/8 (8)
> 5/4 (8)
> 5/3 (8)
> 16/9 (9)
> 9/5 (9)
> 7/4 (10)
> 6/5 (11)
> 10/9 (11)
> 7/5 (12)
> 15/8 (12)
> 16/15 (13)
>
> If I would be a brain I would store ratios in this way....

My ears have some qualms with this ranking.

-Carl

On 03/01/07, Carl Lumma <clumma@yahoo.com> wrote:
>
> Hi Lorenzo,
>
> > Anyway the odd limit is defined on the basis of a prime number
> > which is 2. I think this comes from octave equivalence but it
> > also gives another kind of equivalence which is that of
> > intervals inversions: 4/3 and 3/2 ... have the same odd limit.
>
> 4/3 and 3/2 would have to be the same in any octave-equivalent
> regime, wouldn't they? So this isn't another kind of equivalence,
> it's a facet of octave equivalence.

No. You can have octave equivalence without inversional equivalence.
It's how harmony was explained in the counterpoint books I read. If
you have a harmonic interval larger than an octave, you can make it
smaller by octaves until it's within an octave. Then you apply the
harmonic rules to it. That's how fourths come to be dissonant, but
you don' t have to list the dissonance for intervals larger than an
octave.

It's difficult to state mathematically. You always have to measure
from the lower note up to the higher one. That's similar to giving a
direction to the interval. So for a C-F fourth you aren't allowed to
lower the F another octave and make it an F-C fifth. An
octave-equivalent group of intervals can distinguish directions but
not inversions as such. Another way of looking at it is you can
enforce an octave equivalence of intervals but not of pitches.

Graham

Hiya Graham - good to hear from you.

> You can have octave equivalence without inversional equivalence.
> It's how harmony was explained in the counterpoint books I read. If
> you have a harmonic interval larger than an octave, you can make it
> smaller by octaves until it's within an octave.

If I have octave equivalence, I feel like I should be able
to play F-C, and then move C down an octave. Octave equivalence
is typically drawn on *pitches*, not intervals. But sure, I
guess either way is valid.

> Then you apply the
> harmonic rules to it. That's how fourths come to be dissonant,

I don't think fourths became dissonant for their sound in
naked dyads. It was instead a result of functional harmony.
Perhaps that's what you meant.

> It's difficult to state mathematically. You always have to measure
> from the lower note up to the higher one. That's similar to giving
> a direction to the interval.

Yup, seems unnatural.

> So for a C-F fourth you aren't allowed to
> lower the F another octave and make it an F-C fifth.

Hey, I just wrote that!

> Another way of looking at it is you can
> enforce an octave equivalence of intervals but not of pitches.

That's backward I think.

-Carl

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

> 4/3 and 3/2 would have to be the same in any octave-equivalent
> regime, wouldn't they? So this isn't another kind of equivalence,
> it's a facet of octave equivalence.

In the approach I take, as far as consonance/dissonance, 4/3
belongs to the family of fifths and and 3/2, octaves.

> > Why is this done only for number 2?
>
> Because psychoacoustics tells us that 2 is special.

Plain old acoustics, too, just plot out some overtones of two
pitches of an n/2^ interval relationship. It can be like a filter
applied to a single tone.

> The high degree of symmetry at 3/2 in the world's scales suggests
> something like '3/2-equivalence' is going on.

I agree, and listen to some Georgian (Gruzija) folk music.

> > In any case is really hard to my ears to accept that 8/7 sounds
> > smoother than 9/4 or 16/9
>
> The "critical band" must also be considered. 8/7 is pretty
> close to it, and is within it for much of the musical
> pitch range.

My approach explains this by considering 8/7 part of the seven
familiy, and a very strong one at that. In relation to a 1/1, 8/7 is
also telling us about 7/4, which may be the most strong-willed
harmonic of all. Spend some time in additive synthesis and hear for
yourself what a meaty thing 7 is.
>
> > to have a good ranking for consonance we have to consider that
> > for odd/even ratios the consonance is in inverse relation to
> > n*d or n+d while for even/odd ratios is the opposite: try
> > comparing 12/7 with 8/7,
>
> I will (and get back to you).

That's an interesting idea... have to check it out, too.

> Prime limit alone leads to weird results,
> since there are 3-limit commas arbitrarily close to any
> interval you care to name.

Yes, that's what has bothered me about the idea for a long time.

> > or maybe
> > because each odd limit has a threshold of n*d where it begins
> > to be worst than the successive limit.
>
> If you could quantify this suggestion...

Whooo, to quantify something like this exactly... I think it works
in slightly fuzzy zones.

> > I don't know if there is some relation
> > between primes and perception (any papers?)
>
> I suspect there is some sort of factoring going on in the
> auditory system -- a hierarchy of periodicities, perhaps --
> but if so it doesn't go very far out into the number system,
> and is generally overwhelmed by other facets of hearing like
> roughness and interval size.

I think it's like a decathalon and ballroom dancing put together.

-Cameron Bobro

> > I suspect there is some sort of factoring going on in the
> > auditory system -- a hierarchy of periodicities, perhaps --
> > but if so it doesn't go very far out into the number system,
> > and is generally overwhelmed by other facets of hearing like
> > roughness and interval size.
>
> I think it's like a decathalon and ballroom dancing put together.

I like it!

-Carl

I promised...

> > to have a good ranking for consonance we have to consider that
> > for odd/even ratios the consonance is in inverse relation to
> > n*d or n+d while for even/odd ratios is the opposite: try
> > comparing 12/7 with 8/7,
>
> I will (and get back to you).
>
> > 12/5 with 8/5,
>
> Interval size comes into play here.
>
> > 16/9 with 14/9 and this with 10/9.
>
> 10/9 has even more critical band issues than 8/7.

I plugged 8/7, 14/9, 12/7, and 16/9 into Scala, and I
have to say my hearing agrees perfectly with n*d.

Also, 1/1-8/7-12/7 and 1/1-14/9-16/9 are cool triads!

I should say I compared to the dyads with their lower
notes sharing a single pitch. Probably one should also
compare them with the upper pitch fixed...

-Carl

>> You can have octave equivalence without inversional equivalence.
>> It's how harmony was explained in the counterpoint books I read. If
>> you have a harmonic interval larger than an octave, you can make it
>> smaller by octaves until it's within an octave.
>
>If I have octave equivalence, I feel like I should be able
>to play F-C, and then move C down an octave. Octave equivalence
>is typically drawn on *pitches*, not intervals. But sure, I
>guess either way is valid.

I think that "octave equivalence" is a contraddiction in terms. Each
musician knows that everything in different octaves sounds very different
even if
the name of notes is the same. Even mathematically, for n><0, is absurd to
say that k =
k(12n). So I would prefer to call it "octave similitude".

Inversions are just a specular symmetry applied to the octave interval.
There is no equivalence in it, just a symmetry based on a similitude.

I don't see any reason to restrict this symmetry only to the octave while it
seems that the odd limit does it.

I was wrong in the example I did previously: it's not correct to say that
6/5 is specular to 9/5 respect 3/2. While is correct to do inversions inside
the same interval of fifth. For example 6/5 ~ 5/4, 4/3 ~ 9/8, 45/32 ~ 16/15,
where
"~" means "similar". This could be described as a specular symmetry based on
a "fifth similitude". You can also do the same with "fourth similitude":
10/9 ~ 9/5, 5/4 ~ 16/15, 7/6 ~ 8/7.

These two types of symmetry are also at the origin of parallel scales
similitude:
major scale (2 2 1 2 | 2 2 1)
minor scale (2 1 2 2 | 1 2 2)

It's possible to go on applying specular symmetry on "major third
similitude": 9/8 ~ 10/9, 6/5 ~ 25/24 and "minor third similitude": 9/8 ~
16/15 but it's clear that while octave similitude is the strongest of all,
fifth and fourth similitudes are weak and thirds similitude is almost
nothing.

>Incidentally, Paul Erlich showed that if you start with n*d
>and add only octave equivalence, you get odd limit.

I don't understand this. Can you explain better?

>> In any case is really hard to my ears to accept that 8/7 sounds
>> smoother than 9/4 or 16/9
>
>The "critical band" must also be considered. 8/7 is pretty
>close to it, and is within it for much of the musical
>pitch range.

I've thought that odd limit was exactly a way to measure the effect of
critical band.

>Prime limit alone leads to weird results,
>since there are 3-limit commas arbitrarily close to any
>interval you care to name.

I don't understand this.

>> because each odd limit has a threshold of n*d where it begins
>> to be worst than the successive limit.
>
>If you could quantify this suggestion...

It's hard since I'm still not convinced about the concept of limit: let's
take 13/7 and 13/8. They have the same prime and odd limit. Why 13/8 sounds
better? Even looking at n*d should'nt be so.

>My ears have some qualms with this ranking.

I'm agree..:-) But it remains the best way to compress ratios. Maybe it
could be useful for some other thing than ranking consonance.

lorenzo

>I plugged 8/7, 14/9, 12/7, and 16/9 into Scala, and I
>have to say my hearing agrees perfectly with n*d.

Do you mean that 8/7 sounds smoother than 16/9?
And that 14/9 sounds better than 16/9?

>Also, 1/1-8/7-12/7 and 1/1-14/9-16/9 are cool triads!

Yes!

lorenzo

> >I plugged 8/7, 14/9, 12/7, and 16/9 into Scala, and I
> >have to say my hearing agrees perfectly with n*d.
>
> Do you mean that 8/7 sounds smoother than 16/9?
> And that 14/9 sounds better than 16/9?

Yes. I take it they don't to you. Have you tried
different timbers or registrations?

-Carl

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@...> wrote:

> >Prime limit alone leads to weird results,
> >since there are 3-limit commas arbitrarily close to any
> >interval you care to name.
>
> I don't understand this.

The 3-limit rationals, like the rationals, are a dense set. Hence given
any positive real number r, and any positive e, we can find an infinity
of numbers of the form q = 2^a 3^b with |r-q| < e. Equivalently, we can
look at it logarithmically (centwise), |cents(r) - cents(q)| < c.

Hi Carl.

>> >I plugged 8/7, 14/9, 12/7, and 16/9 into Scala, and I
>> >have to say my hearing agrees perfectly with n*d.
>> >> Do you mean that 8/7 sounds smoother than 16/9?
>> And that 14/9 sounds better than 16/9?
>
>Yes. I take it they don't to you. Have you tried
>different timbers or registrations?

I've tried with a really open mind... But with a lot of timbres I always hear a bad 8/7 in comparison with 16/9.
Maybe you can suggest me the right timbre or you can upload a wav file.
It would be great to have the opinion of other people about it.

lorenzo

Lorenzo Frizzera wrote:
> I think that "octave equivalence" is a contraddiction in terms. Each
> musician knows that everything in different octaves sounds very different
> even if
> the name of notes is the same. Even mathematically, for n><0, is absurd to
> say that k =
> k(12n). So I would prefer to call it "octave similitude".

This may seem strange, but to me the same note in another octave sounds more similar than any other note in the same octave, like a different shade of the same color. Until you get above the range of the piano (where notes start sounding flatter than they actually are before they lose all sense of pitch entirely), an F in one octave sounds pretty much like the F in the next higher or lower octave, with a difference that's only a little bit more than a difference in timbre.

When you get two notes together, of course, it's easier to notice the difference between a fifth and a fourth, or a third and a sixth, but there's still something very similar in the "flavor" of fourths and fifths, or a third compared with its inversion. I don't know if I'd call it "equivalence", but the octave is pretty much unique in that respect. You can take the highest note in a 3-note chord and transpose it down an octave without much change in the sound of the chord. I don't hear this kind of similarity in tunings that repeat at the 3:1 or any other interval.

But once you get more than two or three notes in a chord, the spacing makes a difference. You might end up with lots of discordant intervals (like minor seconds) if you transpose all the notes into a single octave, while if they're spaced widely enough apart, the chord as a whole is more consonant. So there's a limit to this whole "octave equivalence" idea, but it's more than just an arbitrary mathematical idea.

> >Incidentally, Paul Erlich showed that if you start with n*d
> >and add only octave equivalence, you get odd limit.
>
> I don't understand this. Can you explain better?

Actually he showed that harmonic entropy plus octave equivalence
agrees with odd limit for rational intervals, and pure harmonic
entropy agrees with n*d for rational intervals.

> >> In any case is really hard to my ears to accept that 8/7
> >> sounds smoother than 9/4 or 16/9
> >
> >The "critical band" must also be considered. 8/7 is pretty
> >close to it, and is within it for much of the musical
> >pitch range.
>
> I've thought that odd limit was exactly a way to measure the
> effect of critical band.

Odd limit is a handy way to rank rational intervals by
dissonance. Critical bandwidth effects are one source
of dissonance.

> >Prime limit alone leads to weird results,
> >since there are 3-limit commas arbitrarily close to any
> >interval you care to name.
>
> I don't understand this.

I think Gene took care of this one.

> >My ears have some qualms with this ranking.
>
> I'm agree..:-) But it remains the best way to compress ratios.

How is it better than just prime-factoring the ratios,
eg 5/4 = [-2 0 1]?

-Carl

>Odd limit is a handy way to rank rational intervals by
>dissonance. Critical bandwidth effects are one source
>of dissonance.

What are the other causes?

>The 3-limit rationals, like the rationals, are a dense set. Hence given
>any positive real number r, and any positive e, we can find an infinity
>of numbers of the form q = 2^a 3^b with |r-q| < e. Equivalently, we can
>look at it logarithmically (centwise), |cents(r) - cents(q)| < c.

Since, excluding 2, primes are a subset of odds, I would say that odds are
even worst from this point of view since they are much more than primes.

>How is it better than just prime-factoring the ratios,
>eg 5/4 = [-2 0 1]?

If you have to give this information to a computer you need also to include
the number of units necessary to build each prime number.

>different shade of the same color.
>a little bit more than a difference in timbre.
>I don't know if I'd call it "equivalence",

I'm agree.

>octave is pretty much unique in that respect.
>So there's a limit to this whole "octave
>equivalence" idea, but it's more than just an arbitrary mathematical idea.

I'm agree.

>I don't hear this kind of similarity in tunings that repeat at the 3:1 or
>any other interval.

Me too. I think it is a different kind of similarity.

>You can take the highest note in a 3-note chord and transpose it down an
>octave without much change in the sound of the chord.

This statement, which is now obvious to us, is the result of a long historical process.
I'm not sure but it was Rameau with his "Trait� de l'harmonie" (1722) which was the first to define this.

lorenzo

> >Odd limit is a handy way to rank rational intervals by
> >dissonance. Critical bandwidth effects are one source
> >of dissonance.
>
> What are the other causes?

Unclear. But I've never seen it explained via the critical
band why utonal chords are more dissonant than otonal ones.

> >The 3-limit rationals, like the rationals, are a dense set.
>
> Since, excluding 2, primes are a subset of odds, I would say
> that odds are even worst from this point of view since they
> are much more than primes.

Neither the odd numbers or the primes are dense. The
prime-limit *rationals* are dense.

> >How is it better than just prime-factoring the ratios,
> >eg 5/4 = [-2 0 1]?
>
> If you have to give this information to a computer you need
> also to include the number of units necessary to build each
> prime number.

Not sure what you mean here...

-Carl

> > >The 3-limit rationals, like the rationals, are a dense set.
> >
> > Since, excluding 2, primes are a subset of odds, I would say
> > that odds are even worst from this point of view since they
> > are much more than primes.
>
> Neither the odd numbers or the primes are dense. The
> prime-limit *rationals* are dense.

But the odd-limit rationals aren't. For example, there is
no 3-odd-limit number within 50 cents of 300 cents. But
there is a 3-prime-limit number within 50 cents of 300
cents. And there's one within 0.000001 cents of 300 cents.

-Carl

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@...> wrote:

>
> I think that "octave equivalence" is a contraddiction in terms.
Each
> musician knows that everything in different octaves sounds very
different
> even if
> the name of notes is the same. Even mathematically, for n><0, is
absurd to
> say that k =
> k(12n). So I would prefer to call it "octave similitude".

I agree. And the older and (hopefully) more experienced I get, the
more important I think this is.
>
> Inversions are just a specular symmetry applied to the octave
interval.
> There is no equivalence in it, just a symmetry based on a
similitude.

Symmetries- and harmonies, I think. I believe that when my son as a
tiny baby sang along exactly 2 octaves above, it's a matter of
harmony, not equivalence. The old "perfect concord" term is good.
And our abilitiy to spot symmetries absorbs inversions as natural
phenomena, but they're not "the same".

> I don't see any reason to restrict this symmetry only to the
>octave while it
> seems that the odd limit does it.

I agree, and I think Carl does too- symmetrical tetrachords agree,
too, I'd say.

> >> In any case is really hard to my ears to accept that 8/7 sounds
> >> smoother than 9/4 or 16/9

9/4 is the 9th harmonic of the double octave, 16/9 an octave of
the "fifth-of-the-fifth": they're both related directly to the first
3 overtones, while 8/7 is establishing the 7th overtone.

Try this, if you have Scala: open the tuning blackjack_r.scl
("Wilson/Grady", a sweet tuning by the way) and don't click "show".
Open the virtual keyboard and noodle around a bit- what is the one
interval that sticks out? Then check out its ratio. (I did this by
accident today).

> It's hard since I'm still not convinced about the concept of
>limit: let's
> take 13/7 and 13/8. They have the same prime and odd limit. Why
>13/8 sounds
> better? Even looking at n*d should'nt be so.

See above..

>> Neither the odd numbers or the primes are dense. The
>> prime-limit *rationals* are dense.
>
>But the odd-limit rationals aren't. For example, there is
>no 3-odd-limit number within 50 cents of 300 cents. But
>there is a 3-prime-limit number within 50 cents of 300
>cents. And there's one within 0.000001 cents of 300 cents.

Now I have understood. Thanks!

>9/4 is the 9th harmonic of the double octave, 16/9 an octave of
>the "fifth-of-the-fifth": they're both related directly to the first
>3 overtones, while 8/7 is establishing the 7th overtone.

I agree and my ears too. Would this mean that the prime limit has a role in ranking consonance?

>Try this, if you have Scala: open the tuning blackjack_r.scl
>("Wilson/Grady", a sweet tuning by the way) and don't click "show".
>Open the virtual keyboard and noodle around a bit- what is the one
>interval that sticks out? Then check out its ratio. (I did this by
>accident today).

I've played without knowing the ratios of each note. In the first minute of playing I noticed 18/11. But after a while, listening better to the smoothness of each dyads (with 1/1 as the lower note), I've seen that my ears perfectly agree with odd limit since my worst dyads was: 11/9 18/11, 21/16, 32/21, 63/32.
My question is: what I noticed at the beginning?

Another question is about p/n where p is prime and n is a natural numbers minor than p. Ranking all these ratios there should be a constant increasing of dissonance for example with 19/1 19/2 19/3 19/4 ... 19/18.
This does'nt happen with any timbre I've tried. For each timbre there is a sort of ondulation in ranking smoothness. And for each timbre this ondulation is different (for some timbre 19/11 sounds better than 19/10, for others 19/10 is better than 19/9, ...). Do exists any timbre which perfectly agree with odd limit in this context? I have not tried with sine complex tones but with "Microsoft GS wavetable" and I understand that I could have been a big limit... :)

The last question is: why should we consider odd limit instead of n-limit where n is the greatest number of the ratio (example 18/7 is 18-limit instead of 7-limit)?

lorenzo

Hi Lorenzo,

> >> Neither the odd numbers or the primes are dense. The
> >> prime-limit *rationals* are dense.
> >
> >But the odd-limit rationals aren't. For example, there is
> >no 3-odd-limit number within 50 cents of 300 cents. But
> >there is a 3-prime-limit number within 50 cents of 300
> >cents. And there's one within 0.000001 cents of 300 cents.
>
> Now I have understood. Thanks!

Sure!

> >9/4 is the 9th harmonic of the double octave, 16/9 an octave of
> >the "fifth-of-the-fifth": they're both related directly to the
> >first 3 overtones, while 8/7 is establishing the 7th overtone.
>
> I agree and my ears too. Would this mean that the prime limit
> has a role in ranking consonance?

It may in some way, but no suggestion I'm aware of has ever
done better than n*d at ranking ratios in tests I've tried.

> I've played without knowing the ratios of each note. In the
> first minute of playing I noticed 18/11. But after a while,
> listening better to the smoothness of each dyads (with 1/1
> as the lower note), I've seen that my ears perfectly agree
> with odd limit since my worst dyads was: 11/9 18/11,
> 21/16, 32/21, 63/32.

Great!

> My question is: what I noticed at the beginning?

I don't know. Keep exploring, and if you figure it out
let us know... and tell us how to hear it too. :)

> Another question is about p/n where p is prime and n is
> a natural numbers minor than p. Ranking all these ratios
> there should be a constant increasing of dissonance for
> example with 19/1 19/2 19/3 19/4 ... 19/18.
> This does'nt happen with any timbre I've tried. For each
> timbre there is a sort of ondulation in ranking smoothness.
> And for each timbre this ondulation is different (for some
> timbre 19/11 sounds better than 19/10, for others 19/10 is
> better than 19/9, ...). Do exists any timbre which perfectly
> agree with odd limit in this context? I have not tried with
> sine complex tones but with "Microsoft GS wavetable" and I
> understand that I could have been a big limit... :)

Odd limit doesn't deny there are variations between such
ratios, it merely says these variations should be smaller
than the difference between any one of them and any ratio
like 17/x. But the whole thing only works for relatively
small numbers. By the time the 19-limit is reached, you
can approximate other intervals... for example 13/11 and
19/16 and 6/5 all seem to sound like a "minor third", and
I don't think this purely a result of our cultural bias.

Another thing is that 19/1 is a huge interval, and as
intervals get very large, their consonance AND their
dissonance decreases. They tones cease to interact in
either way.

> The last question is: why should we consider odd limit
> instead of n-limit where n is the greatest number of the
> ratio (example 18/7 is 18-limit instead of 7-limit)?

The reason to use odd limit is when octave equivalence
is desired. If you don't want octave equivalence, n*d
is better. In that case 18/7 will have a higher
dissonance rating than many other 7-limit ratios like
7/4 or even 10/7.

-Carl

I always like to chime in on discussions like this,
because what you two are saying supports my ideas about
the prime-factors of the ratios (or perceived ratios)
being the important _gestalt_ of the sound of an interval.

According to my theory (and it didn't start with me,
cf. Helmholtz, Partch, etc.), the "special" similitude
exhibited by the "octave" is simply the affect of
prime-factor 2.

Prime-factor 3 has its own unique affect, which was
exploited abundantly in European church music during
the Medieval period, and 5 has its affect, which began
to figure prominently around 1500, and 7 has its affect,
which really came to the fore in barbershop and the blues,
etc. etc.

See the Tonalsoft Encyclopedia entry of "affect":
http://tonalsoft.com/enc/a/affect.aspx

The most comprehensive and interesting articles i've seen
about prime-affect were those written by Scott Makeig
in various publications, including _Interval_ (and IIRC
_Xenharmonikon_).

Regarding the question of why 13/8 "sounds better"
(which is a matter of opinion anyway) than 13/7, i
would say that it's because of sizes of the primes
involved in the ratios: 2 & 13 indicates something
more concordant to me than 7 & 13.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Lorenzo Frizzera"
> <lorenzo.frizzera@> wrote:
>
> >
> > I think that "octave equivalence" is a contraddiction
> > in terms. Each musician knows that everything in different
> > octaves sounds very different even if the name of notes
> > is the same. Even mathematically, for n><0, is absurd to
> > say that k = k(12n). So I would prefer to call it
> > "octave similitude".
>
> I agree. And the older and (hopefully) more experienced I get,
> the more important I think this is.
> >
> > Inversions are just a specular symmetry applied to the
> > octave interval.
> > There is no equivalence in it, just a symmetry based on
> > a similitude.
>
> Symmetries- and harmonies, I think. I believe that when my son as a
> tiny baby sang along exactly 2 octaves above, it's a matter of
> harmony, not equivalence. The old "perfect concord" term is good.
> And our abilitiy to spot symmetries absorbs inversions as natural
> phenomena, but they're not "the same".
>
> > I don't see any reason to restrict this symmetry only to
> > the octave while it seems that the odd limit does it.
>
> I agree, and I think Carl does too- symmetrical tetrachords agree,
> too, I'd say.
>
> > >> In any case is really hard to my ears to accept that 8/7 sounds
> > >> smoother than 9/4 or 16/9
>
> 9/4 is the 9th harmonic of the double octave, 16/9 an octave of
> the "fifth-of-the-fifth": they're both related directly to the first
> 3 overtones, while 8/7 is establishing the 7th overtone.
>
> Try this, if you have Scala: open the tuning blackjack_r.scl
> ("Wilson/Grady", a sweet tuning by the way) and don't click "show".
> Open the virtual keyboard and noodle around a bit- what is the one
> interval that sticks out? Then check out its ratio. (I did this by
> accident today).
>
>
> > It's hard since I'm still not convinced about the concept
> > of limit: let's take 13/7 and 13/8. They have the same prime
> > and odd limit. Why 13/8 sounds better? Even looking at n*d
> > should'nt be so.
>
> See above..

> I always like to chime in on discussions like this,
> because what you two are saying supports my ideas about
> the prime-factors of the ratios (or perceived ratios)
> being the important _gestalt_ of the sound of an interval.
>
> According to my theory (and it didn't start with me,
> cf. Helmholtz, Partch, etc.), the "special" similitude
> exhibited by the "octave" is simply the affect of
> prime-factor 2.

In what way did Partch or Helmholtz say... what exactly?

> Prime-factor 3 has its own unique affect, which was
> exploited abundantly in European church music during
> the Medieval period, and 5 has its affect, which began
> to figure prominently around 1500, and 7 has its affect,
> which really came to the fore in barbershop and the blues,
> etc. etc.

How far out does this go? Is 27/16 have a 3-affect?
Does 19683/16384?

> Regarding the question of why 13/8 "sounds better"
> (which is a matter of opinion anyway) than 13/7,

What's your opinion?

> i would say that it's because of sizes of the primes
> involved in the ratios: 2 & 13 indicates something
> more concordant to me than 7 & 13.

What about 13/11?

-Carl

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@...> wrote:

>
> >9/4 is the 9th harmonic of the double octave, 16/9 an octave of
> >the "fifth-of-the-fifth": they're both related directly to the
first
> >3 overtones, while 8/7 is establishing the 7th overtone.
>
> I agree and my ears too. Would this mean that the prime limit has
>a role in
> ranking consonance?

Well I believe so- but not by virtue of some magic power of prime
numbers, but by how it relates to overtones. Odd and even numbers
would then also have roles, first of all because so many even
numbers relate to the octave family (in the way I percieve it).

> >Try this, if you have Scala: open the tuning blackjack_r.scl
> >("Wilson/Grady", a sweet tuning by the way) and don't
>click "show".
> >Open the virtual keyboard and noodle around a bit- what is the one
> >interval that sticks out? Then check out its ratio. (I did this by
> >accident today).
>
> I've played without knowing the ratios of each note. In the first
>minute of playing I noticed 18/11.

Exactly, same here. :-)

But after a while, listening better to the
> smoothness of each dyads (with 1/1 as the lower note), I've seen
>that my
> ears perfectly agree with odd limit since my worst dyads was: 11/9
>18/11,
> 21/16, 32/21, 63/32.

Pretty much the same here, although I wouldn't call the
intervals "worse", but simply different ( and vive la difference!)

> My question is: what I noticed at the beginning?

Well according to my theory, we both noticed the interval that
belongs to a different harmonic family than all the rest.

> Another question is about p/n where p is prime and n is a natural
>numbers
> minor than p. Ranking all these ratios there should be a constant
>increasing
> of dissonance for example with 19/1 19/2 19/3 19/4 ... 19/18.
> This does'nt happen with any timbre I've tried. For each timbre
>there is a
> sort of ondulation in ranking smoothness.

I agree completely and this is the point of the Csound interval
testing code I posted here- in that example there is, to my ears,
and undulation of consonance and a strong example of a dip, or jump,
in consonance in the intervals as they move away from 3/2.

>And for each timbre this
> ondulation is different (for some timbre 19/11 sounds better than
>19/10, for
> others 19/10 is better than 19/9, ...). Do exists any timbre which
>perfectly
> agree with odd limit in this context? I have not tried with sine
>complex
> tones but with "Microsoft GS wavetable" and I understand that I
>could have
> been a big limit... :)

ZynAddSubFX is a freeware softsynth that takes Scala tuning files
and keyboard mappings. It is as accurate as any sane person could
desire- I've tested it against the most precision electronic tuning
method I know of, which is calculating freqencies directly by ratio
in 64-bit Csound (cycles per second equals base frequency multiplied
by ratio). In ZynAddSubFX, there are additive synthesizers that let
you change the amplitude of the first 64 partials, and you can hear
as plain as day, in real time, how important timbre is for tuning.
Try it!

> The last question is: why should we consider odd limit instead of
>n-limit
> where n is the greatest number of the ratio (example 18/7 is 18-
>limit
> instead of 7-limit)?

For me it's not the "limit", at least not in the more or
less "simple" ratios I work with (864/275 is an example of how about
how complex they get). Limits may come in somewhere above that
point, don't know. It's the character, or "affect" as Monz just
pointed out, of the harmonic (denominator). The strength of that
affect would then be modified by the numerator. In this case, 18/7,
the 7-family affect would not really be weakened or soured by the 9-
family overtone above it. In the case of, say, 11/7, you've got a
strong family, 7, and it's being described so to speak by another
strong family, 11, and (according to my theory), you would expect
something strange and powerful in relation to the 1/1, but not
weird, because the whole shebang isn't THAT far removed as far as
overtones.

-Cameron Bobro

"Affect" is the perfect word! The best I could think of
was "Wirkung", and I didn't want to use German.

As far as primes, a great number of the most audible partials
are primes, and I think it may never be possible to actually
know whether the "family affects" I've been talking about arise
because of prime numbers, or because we've got such a collection
of different character families right in front of us which
happen to be founded on primes.

The problem is, IMO, that whether we use primes or odd numbers,
or simply sheer bulk of number or ratio, we rocket off pretty
quickly into such distant overtones that I cannot "prove" to myself
their character with additive synthesis, and soon after that
we're inan esoteric zone where we could only debate about the
relevance of things which are inaudible on a conscious level.

-Cameron Bobro

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> I always like to chime in on discussions like this,
> because what you two are saying supports my ideas about
> the prime-factors of the ratios (or perceived ratios)
> being the important _gestalt_ of the sound of an interval.
>
> According to my theory (and it didn't start with me,
> cf. Helmholtz, Partch, etc.), the "special" similitude
> exhibited by the "octave" is simply the affect of
> prime-factor 2.
>
> Prime-factor 3 has its own unique affect, which was
> exploited abundantly in European church music during
> the Medieval period, and 5 has its affect, which began
> to figure prominently around 1500, and 7 has its affect,
> which really came to the fore in barbershop and the blues,
> etc. etc.
>
>
> See the Tonalsoft Encyclopedia entry of "affect":
> http://tonalsoft.com/enc/a/affect.aspx
>
> The most comprehensive and interesting articles i've seen
> about prime-affect were those written by Scott Makeig
> in various publications, including _Interval_ (and IIRC
> _Xenharmonikon_).
>
>
> Regarding the question of why 13/8 "sounds better"
> (which is a matter of opinion anyway) than 13/7, i
> would say that it's because of sizes of the primes
> involved in the ratios: 2 & 13 indicates something
> more concordant to me than 7 & 13.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> >
> > --- In tuning@yahoogroups.com, "Lorenzo Frizzera"
> > <lorenzo.frizzera@> wrote:
> >
> > >
> > > I think that "octave equivalence" is a contraddiction
> > > in terms. Each musician knows that everything in different
> > > octaves sounds very different even if the name of notes
> > > is the same. Even mathematically, for n><0, is absurd to
> > > say that k = k(12n). So I would prefer to call it
> > > "octave similitude".
> >
> > I agree. And the older and (hopefully) more experienced I get,
> > the more important I think this is.
> > >
> > > Inversions are just a specular symmetry applied to the
> > > octave interval.
> > > There is no equivalence in it, just a symmetry based on
> > > a similitude.
> >
> > Symmetries- and harmonies, I think. I believe that when my son
as a
> > tiny baby sang along exactly 2 octaves above, it's a matter of
> > harmony, not equivalence. The old "perfect concord" term is
good.
> > And our abilitiy to spot symmetries absorbs inversions as
natural
> > phenomena, but they're not "the same".
> >
> > > I don't see any reason to restrict this symmetry only to
> > > the octave while it seems that the odd limit does it.
> >
> > I agree, and I think Carl does too- symmetrical tetrachords
agree,
> > too, I'd say.
> >
> > > >> In any case is really hard to my ears to accept that 8/7
sounds
> > > >> smoother than 9/4 or 16/9
> >
> > 9/4 is the 9th harmonic of the double octave, 16/9 an octave of
> > the "fifth-of-the-fifth": they're both related directly to the
first
> > 3 overtones, while 8/7 is establishing the 7th overtone.
> >
> > Try this, if you have Scala: open the tuning blackjack_r.scl
> > ("Wilson/Grady", a sweet tuning by the way) and don't
click "show".
> > Open the virtual keyboard and noodle around a bit- what is the
one
> > interval that sticks out? Then check out its ratio. (I did this
by
> > accident today).
> >
> >
> > > It's hard since I'm still not convinced about the concept
> > > of limit: let's take 13/7 and 13/8. They have the same prime
> > > and odd limit. Why 13/8 sounds better? Even looking at n*d
> > > should'nt be so.
> >
> > See above..
>

Can you tell me how to set it up so that it works in Windows? I don't
understand a thing about compilation!

Oz.

SNIP

>
> ZynAddSubFX is a freeware softsynth that takes Scala tuning files
> and keyboard mappings. It is as accurate as any sane person could
> desire- I've tested it against the most precision electronic tuning
> method I know of, which is calculating freqencies directly by ratio
> in 64-bit Csound (cycles per second equals base frequency multiplied
> by ratio). In ZynAddSubFX, there are additive synthesizers that let
> you change the amplitude of the first 64 partials, and you can hear
> as plain as day, in real time, how important timbre is for tuning.
> Try it!
>

SNIP

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Can you tell me how to set it up so that it works in Windows? I don't
> understand a thing about compilation!
>
> Oz.
>

There's a Windows port, thank goodness! I got it from another site
which I now can't find. Anyway it seems to work perfectly, on XP.

http://www.softpedia.com/get/Multimedia/Audio/Audio-Mixers-
Synthesizers/ZynAddSubFX.shtml

Wow, what a great synth! Pity there are so little instruments to go with it.
Pitier still, that I cannot overcome the sound delay between pressing a key
and hearing its associated pitch...

Oz.

----- Original Message -----
From: "Cameron Bobro" <misterbobro@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 06 Ocak 2007 Cumartesi 21:26
Subject: [tuning] Re: Odd limit

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Can you tell me how to set it up so that it works in Windows? I don't
> > understand a thing about compilation!
> >
> > Oz.
> >
>
> There's a Windows port, thank goodness! I got it from another site
> which I now can't find. Anyway it seems to work perfectly, on XP.
>
>
> http://www.softpedia.com/get/Multimedia/Audio/Audio-Mixers-
> Synthesizers/ZynAddSubFX.shtml
>
>

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> The problem is, IMO, that whether we use primes or odd numbers,
> or simply sheer bulk of number or ratio,

I've seen you guys, in several messages in the "Odd limit" thread,
struggling to describe this "sheer bulk of number or ratio".
We've referred to it here in the past as "integer-limit".

(Don't use the hyphens if that's your preference ... i happen
to like them a lot.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > I always like to chime in on discussions like this,
> > because what you two are saying supports my ideas about
> > the prime-factors of the ratios (or perceived ratios)
> > being the important _gestalt_ of the sound of an interval.
> >
> > According to my theory (and it didn't start with me,
> > cf. Helmholtz, Partch, etc.), the "special" similitude
> > exhibited by the "octave" is simply the affect of
> > prime-factor 2.
>
> In what way did Partch or Helmholtz say... what exactly?

Aw, i'd have to dig into the books and look it up, and
can't do that right now (and can't promise when either).
Sorry.

But as for Partch, i can point you to his Three
Observations ... i think they're in the chapter called
"The Question of Resolution". He didn't really talk there
about affect, but rather about the "sphere of influence"
(or some similar term) which is directly proportional
to the ... ah, shucks, i think now i recall that he
said it's directly proportional to the logarithmic
interval size. OK, i could be all wrong about this ...
but Partch's entire theory is in fact founded on the
idea that each prime-factor contributes something
distinctive to the sound of the music/tuning.

> > Prime-factor 3 has its own unique affect, which was
> > exploited abundantly in European church music during
> > the Medieval period, and 5 has its affect, which began
> > to figure prominently around 1500, and 7 has its affect,
> > which really came to the fore in barbershop and the blues,
> > etc. etc.
>
> How far out does this go? Is 27/16 have a 3-affect?
> Does 19683/16384?

In effect, you're asking what is the limit on the *exponents*
of the prime-factors in demonstrating prime-affect.

On my Encyclopedia "affect" page (link again:)

http://tonalsoft.com/enc/a/affect.aspx

I used the disclaimer: "this categorical interval quantization
is based on the easy recognition of small prime-number factors,
up to at least 13, possibly up to 19 or 23, possibly much higher."
So much for the prime-factors.

However, i never did say anything there about how far the
exponents can go and still demonstrate prime-affect, a lack
which i should remedy.

I would say that, similarly, it is only observable for
the lowest few exponents, or perhaps it has something to do
with the integer-limit or odd-limit which is obtained by
the combination of the size of the prime-factor and the size
of its exponent.

But in any case, it's rather low.

So i would never claim that 19683/16384 = 2-3-monzo [-14 9>
exhibits 3-affect -- quite the contrary: it's so close
in size to 6/5 (less than 2 cents larger) that it probably
seems to exhibit 5-affect much more strongly. Empirical
testing is a good idea here: what does everyone out there
have to say about this example?

27/16 = 2-3-monzo [-4 3> i would say is less clear-cut.
This interval, the pythagorean major-6th, is clearly
different from 5/3, the 5-limit-JI major-6th, being
quite a bit larger (a full syntonic-comma, ~21.5 cents),
and also clearly less concordant (or equivalently,
more discordant) than its 5-limit counterpart. I personally
think that 3-limit intervals up to an exponent-limit of 4
(i.e., 81/64 and 128/81), and possibly even 5, exhibit "3-ness",
so i would count 27/16 as exhibiting "3-ness" -- but many
others disagree.

One thing i would *definitely* claim is: as the size of the
prime becomes larger, the exponent-limit which exhibits affect
becomes smaller, so that for 5-limit intervals, perhaps
only 5^1 (5/4 and 8/5) and 5^2 (25/16 and 32/25) exhibit
"5-ness" ... i personally would hesitate to claim that
5^3 still produces the affect. Not that i would definitely
rule it out, but i would hesitate.

And for 7-limit intervals and all prime-factors above 7,
the exponent-limit for affect would only be 1 ... again,
my own personal opinion.

> > Regarding the question of why 13/8 "sounds better"
> > (which is a matter of opinion anyway) than 13/7,
>
> What's your opinion?

I would say that 13/8 is more concordant than 13/7,
for the reason which appears in the next-quoted passage.

But "sounds better" might not mean "more concordant",
which is why i put in that stuff about "opinion".

> > i would say that it's because of sizes of the primes
> > involved in the ratios: 2 & 13 indicates something
> > more concordant to me than 7 & 13.
>
> What about 13/11?

By strict numerical analysis, 13/11 would, according to
my criteria, be less concordant than either 13/8 or 13/7.

However, it is less than 5 cents smaller than 32/27, so
i'd say there's a good chance that 13/11 could be perceived
as 32/27 (depending on musical context, of course, if there
is any, which there wouldn't be in a strict psychoacoustical
experiment). Now, is 32/27 more or less concordant than
13.8 or 13/7? Dunno ... i'll read the oberservations of
others with interest and do my own listening test when i
get a chance.

-monz
http://tonalsoft.com
Tonescape microtonal music software

SNIP

>
> So i would never claim that 19683/16384 = 2-3-monzo [-14 9>
> exhibits 3-affect -- quite the contrary: it's so close
> in size to 6/5 (less than 2 cents larger) that it probably
> seems to exhibit 5-affect much more strongly. Empirical
> testing is a good idea here: what does everyone out there
> have to say about this example?
>

The number of prime exponents in the numerator of the ratio is 9,
denominator is 14. I do not know whether it makes sense at all, but
subtraction of the prime exponent of the latter from the former gives the
number 5. Is this relevant, significant, coincidental?

> 27/16 = 2-3-monzo [-4 3> i would say is less clear-cut.
> This interval, the pythagorean major-6th, is clearly
> different from 5/3, the 5-limit-JI major-6th, being
> quite a bit larger (a full syntonic-comma, ~21.5 cents),
> and also clearly less concordant (or equivalently,
> more discordant) than its 5-limit counterpart. I personally
> think that 3-limit intervals up to an exponent-limit of 4
> (i.e., 81/64 and 128/81), and possibly even 5, exhibit "3-ness",
> so i would count 27/16 as exhibiting "3-ness" -- but many
> others disagree.

Just because it exhibits 3-ness, you categorize it as discordant? Even
though it makes just the right interval in the dominant seventh 3rd
inversion expressed as 54675:48600:34560:20480?

>
> One thing i would *definitely* claim is: as the size of the
> prime becomes larger, the exponent-limit which exhibits affect
> becomes smaller, so that for 5-limit intervals, perhaps
> only 5^1 (5/4 and 8/5) and 5^2 (25/16 and 32/25) exhibit
> "5-ness" ... i personally would hesitate to claim that
> 5^3 still produces the affect. Not that i would definitely
> rule it out, but i would hesitate.
>

What? It most definitely sounds 5-some in the dominant seventh 2nd inversion
expressed as 72:60:45:27.

> And for 7-limit intervals and all prime-factors above 7,
> the exponent-limit for affect would only be 1 ... again,
> my own personal opinion.
>
>

Even though 20250:15750:11025:6174 sounds 7-some?

>
> > > Regarding the question of why 13/8 "sounds better"
> > > (which is a matter of opinion anyway) than 13/7,
> >
> > What's your opinion?
>
> I would say that 13/8 is more concordant than 13/7,
> for the reason which appears in the next-quoted passage.
>
> But "sounds better" might not mean "more concordant",
> which is why i put in that stuff about "opinion".
>
>

Even though it is the sensible (-1th) in this scale?

-2: 27/32 -294.135 Pythagorean minor third
-1: 13/14 -128.298 2/3-tone
0: 1/1 0.000 unison, perfect prime
1: 13/12 138.573 tridecimal 2/3-tone
2: 26/21 369.747
3: 4/3 498.045 perfect fourth
4: 3/2 701.955 perfect fifth
5: 13/8 840.528 tridecimal neutral sixth
6: 13/7 1071.702 16/3-tone
7: 2/1 1200.000 octave

> > > i would say that it's because of sizes of the primes
> > > involved in the ratios: 2 & 13 indicates something
> > > more concordant to me than 7 & 13.
> >
> > What about 13/11?
>
>
> By strict numerical analysis, 13/11 would, according to
> my criteria, be less concordant than either 13/8 or 13/7.
>

Although the scale below sounds excellent as it is, but terrible when 13/8
is swapped for 52/33?

0: 1/1 0.000 unison, perfect prime
1: 12/11 150.637 3/4-tone, undecimal neutral second
2: 13/11 289.210 tridecimal minor third
3: 14/11 417.508 undecimal diminished fourth or major
third
4: 3/2 701.955 perfect fifth
5: 52/33 787.255 tridecimal minor sixth
6: 39/22 991.165
7: 2/1 1200.000 octave

> However, it is less than 5 cents smaller than 32/27, so
> i'd say there's a good chance that 13/11 could be perceived
> as 32/27 (depending on musical context, of course, if there
> is any, which there wouldn't be in a strict psychoacoustical
> experiment). Now, is 32/27 more or less concordant than
> 13.8 or 13/7? Dunno ... i'll read the oberservations of
> others with interest and do my own listening test when i
> get a chance.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

Oz.

I too like hyphenization, but where is this integer-limit in your
monz-opedia monz?

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 07 Ocak 2007 Pazar 5:31
Subject: [tuning] integer-limit (was: Odd limit)

> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> > The problem is, IMO, that whether we use primes or odd numbers,
> > or simply sheer bulk of number or ratio,
>
>
> I've seen you guys, in several messages in the "Odd limit" thread,
> struggling to describe this "sheer bulk of number or ratio".
> We've referred to it here in the past as "integer-limit".
>
> (Don't use the hyphens if that's your preference ... i happen
> to like them a lot.)
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>

> Partch's entire theory is in fact founded on the
> idea that each prime-factor contributes something
> distinctive to the sound of the music/tuning.

Is there any evidence at all to support this statement?

> > How far out does this go? Is 27/16 have a 3-affect?
> > Does 19683/16384?
>
> In effect, you're asking what is the limit on the *exponents*
> of the prime-factors in demonstrating prime-affect.
//
> I used the disclaimer: "this categorical interval quantization
> is based on the easy recognition of small prime-number factors,
> up to at least 13, possibly up to 19 or 23, possibly much
> higher." So much for the prime-factors.

What does '"possibly much higher" than 23' mean?

> However, i never did say anything there about how far the
> exponents can go and still demonstrate prime-affect, a lack
> which i should remedy.
>
> I would say that, similarly, it is only observable for
> the lowest few exponents,
//
> So i would never claim that 19683/16384 = 2-3-monzo [-14 9>
> exhibits 3-affect -- quite the contrary: it's so close
> in size to 6/5 (less than 2 cents larger) that it probably
> seems to exhibit 5-affect much more strongly. Empirical
> testing is a good idea here: what does everyone out there
> have to say about this example?

Sounds like 6/5 to me.

> 27/16 = 2-3-monzo [-4 3> i would say is less clear-cut.
> This interval, the pythagorean major-6th, is clearly
> different from 5/3, the 5-limit-JI major-6th, being
> quite a bit larger (a full syntonic-comma, ~21.5 cents),
> and also clearly less concordant (or equivalently,
> more discordant) than its 5-limit counterpart. I personally
> think that 3-limit intervals up to an exponent-limit of 4
> (i.e., 81/64 and 128/81), and possibly even 5,

Well, you're up to 7 on the 2 part of 127/81... How do I
tell whether the 2-affect or the 3-affect will be dominant?

> > > Regarding the question of why 13/8 "sounds better"
> > > (which is a matter of opinion anyway) than 13/7,
> >
> > What's your opinion?
>
> I would say that 13/8 is more concordant than 13/7,
> for the reason which appears in the next-quoted passage.

To me, 13/7 sounds less discordant, and about as
concordant, than 13/8. If you look at the graph at

/harmonic_entropy

you can see that both have about the same entropy, but
that 13/8 is at a pretty tight local maximum, whereas
13/7 is on a rather broad shelf; a finding which seems
to support my perception.

> > > i would say that it's because of sizes of the primes
> > > involved in the ratios: 2 & 13 indicates something
> > > more concordant to me than 7 & 13.
> >
> > What about 13/11?
>
> By strict numerical analysis, 13/11 would, according to
> my criteria, be less concordant than either 13/8 or 13/7.
>
> However, it is less than 5 cents smaller than 32/27, so
> i'd say there's a good chance that 13/11 could be perceived
> as 32/27 (depending on musical context, of course, if there
> is any, which there wouldn't be in a strict psychoacoustical
> experiment). Now, is 32/27 more or less concordant than
> 13.8 or 13/7? Dunno ... i'll read the oberservations of
> others with interest and do my own listening test when i
> get a chance.

Leaving musical context out of this completely, 32/27 and
13/11 are both in the 'basin of attraction', if you will,
of 6/5, according to the graph on the harmonic_entropy group
home page. So is 19/16, which I think is the most concordant
of the lot, followed by 13/11 followed by 32/27.

Even more convincingly, 27/16 is in the 5/3 bucket. Is it
like a 5/3, but with 3-ness? Maybe.

FWIW, I think

3/2 and 4/3 share a 3-ness ("square")
5/4 and 5/3 share a 5-ness ("sweet")
7/4, 7/6, 8/7, and 7/5 share a 7-ness ("zesty")

but I don't think it has anything to do with primes, and
I think it's completely independent of the concordance
and discordance question.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Partch's entire theory is in fact founded on the
> > idea that each prime-factor contributes something
> > distinctive to the sound of the music/tuning.
>
> Is there any evidence at all to support this statement?

Obviously 2 and 3 together are different than just 2 or just 3, so we
get that far for free. Does 5 add anything new? Unless we are willing
to cheat by such expedients as using 2^13/3^8 in pace of 5/4, I would
say it does; there is a clear distinctive quality the thirds add to the
harmonic mix.

Keep that up for as long as it remains plausible.

> > > Partch's entire theory is in fact founded on the
> > > idea that each prime-factor contributes something
> > > distinctive to the sound of the music/tuning.
> >
> > Is there any evidence at all to support this statement?
>
> Obviously 2 and 3 together are different than just 2 or just 3,
> so we get that far for free. Does 5 add anything new? Unless we
> are willing to cheat by such expedients as using 2^13/3^8 in
> pace of 5/4, I would say it does; there is a clear distinctive
> quality the thirds add to the harmonic mix.
>
> Keep that up for as long as it remains plausible.

What does this have to do with Partch's theory, and how does
it distinguish between primes and odds?

-Carl

I wrote:

> I suspect there is some sort of factoring going on in the
> auditory system -- a hierarchy of periodicities, perhaps --
> but if so it doesn't go very far out into the number system,
> and is generally overwhelmed by other facets of hearing like
> roughness and interval size. I seem to remember finding a
> vaguely-related paper on this once. I'll try to dig it up.

I searched, but I couldn't find anything.

-Carl

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Wow, what a great synth! Pity there are so little instruments to go
with it.

There are three synthesis engines built in- an additive synth, a
subtractive synth, and a "pad synth" which is, technically, a cousin
of the vocoder ((re)synthesis by bandpass filtering). The interface is
somewhat confusing, but an enormous number of sounds, far more than
the presets show, can be made.

Time consuming of course.

> Pitier still, that I cannot overcome the sound delay between
>pressing a key
> and hearing its associated pitch...
>
> Oz.

What soundcard do you have? With say an M-Audio audiocard and a newish
computer, it will play like a hardware synth, "without" latency (ie,
3-6 ms latency).

If your soundcard and computer aren't up to using it in realtime,
it's still great with a midi sequencer, for testing tunings and
scales at the least.

-Cameron Bobro

----- Original Message -----
From: "Cameron Bobro" <misterbobro@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 07 Ocak 2007 Pazar 23:29
Subject: [tuning] Re: Odd limit

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Wow, what a great synth! Pity there are so little instruments to go
> with it.
>
> There are three synthesis engines built in- an additive synth, a
> subtractive synth, and a "pad synth" which is, technically, a cousin
> of the vocoder ((re)synthesis by bandpass filtering). The interface is
> somewhat confusing, but an enormous number of sounds, far more than
> the presets show, can be made.
>
> Time consuming of course.
>

Do we have free instrument samples already made?

> > Pitier still, that I cannot overcome the sound delay between
> >pressing a key
> > and hearing its associated pitch...
> >
> > Oz.
>
> What soundcard do you have? With say an M-Audio audiocard and a newish
> computer, it will play like a hardware synth, "without" latency (ie,
> 3-6 ms latency).
>

I have a Windows ME platform with a SBLive, Yamaha SW1000XG and DSP Factory
2416 audio card. Even the DSP Factory produces latent sounds. I am
considering a Macbook pro with all the latest gear in a year or so.

> If your soundcard and computer aren't up to using it in realtime,
> it's still great with a midi sequencer, for testing tunings and
> scales at the least.
>
> -Cameron Bobro
>
>
>

Thanks for the insider info.
Oz.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Partch's entire theory is in fact founded on the
> > idea that each prime-factor contributes something
> > distinctive to the sound of the music/tuning.
>
> Is there any evidence at all to support this statement?

I haven't read more than a few words of Partch, but when his
instruments are strummed slowly, it sure sounds like a concrete
demonstration of that very statement to me. I could be wrong.

monz:
> > I used the disclaimer: "this categorical interval quantization
> > is based on the easy recognition of small prime-number factors,
> > up to at least 13, possibly up to 19 or 23, possibly much
> > higher." So much for the prime-factors.

lumma:
> What does '"possibly much higher" than 23' mean?

My physical experience "pulling the drawbars" in additive synthesis
has led me to these same numbers. I'd say definitely to 23, "maybe"
to 31 depending on timbre? and then it's either beyond what I've
explored, or seems to duplicate or fall into a character of an
interval within that range (23).

Monz:
> > > > i would say that it's because of sizes of the primes
> > > > involved in the ratios: 2 & 13 indicates something
> > > > more concordant to me than 7 & 13.

My theory says look at the character of the harmonic (denominator)
first, except in the lowest and simplest of intervals, where the
whole shebang falls within audibility and the interval is probably
actually defining a character family.

lumma:
> FWIW, I think
>
> 3/2 and 4/3 share a 3-ness ("square")
> 5/4 and 5/3 share a 5-ness ("sweet")
> 7/4, 7/6, 8/7, and 7/5 share a 7-ness ("zesty")

I agree, see above...
>
> but I don't think it has anything to do with primes,

I suspect that the fact that the odds and primes are pretty much the
same set within the range of most audible harmonics has given them
too much weight.

> and
> I think it's completely independent of the concordance
> and discordance question

I agree, which is why I keep harping on about character families
rather than brute consonance/dissonance, which is so very context-
dependent.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> Do we have free instrument samples already made?

There is a whole bunch of presets, under "banks" IIRC.

> I have a Windows ME platform with a SBLive, Yamaha SW1000XG and DSP
Factory
> 2416 audio card. Even the DSP Factory produces latent sounds. I am
> considering a Macbook pro with all the latest gear in a year or so.

Maybe it's the Windows ME? Because you shouldn't have a problem with
that setup- can you access the latency settings on your soundcards?
They're usually set very high by default.

> > > Partch's entire theory is in fact founded on the
> > > idea that each prime-factor contributes something
> > > distinctive to the sound of the music/tuning.
> >
> > Is there any evidence at all to support this statement?
>
> I haven't read more than a few words of Partch, but when his
> instruments are strummed slowly, it sure sounds like a concrete
> demonstration of that very statement to me. I could be wrong.

How so?

Also, even if it *sounded* that way, it wouldn't establish
anything about Partch's *theory*.

> monz:
> > > I used the disclaimer: "this categorical interval quantization
> > > is based on the easy recognition of small prime-number factors,
> > > up to at least 13, possibly up to 19 or 23, possibly much
> > > higher." So much for the prime-factors.
>
> lumma:
> > What does '"possibly much higher" than 23' mean?
>
> My physical experience "pulling the drawbars" in additive synthesis
> has led me to these same numbers.

You mean you've found the 23rd partial is influential in
shaping timbres, or the 23rd prime number is influential in
the "affect" of dyads?

> Monz:
> > > > > i would say that it's because of sizes of the primes
> > > > > involved in the ratios: 2 & 13 indicates something
> > > > > more concordant to me than 7 & 13.
>
> My theory says look at the character of the harmonic (denominator)
> first, except in the lowest and simplest of intervals, where the
> whole shebang falls within audibility and the interval is probably
> actually defining a character family.

Everybody's got a theory. Theories must make predictions.
Exactly what are your theories predicting?

> > but I don't think it has anything to do with primes,
>
> I suspect that the fact that the odds and primes are pretty
> much the same set within the range of most audible harmonics
> has given them too much weight.

Below 23 there are three composite odds, and besides, in
the context of monz's posts "primes" implies *factoring*.
It's a shame that Paul Erlich so patiently showed, over the
span of a decade on this list, that such claims are
groundless, only to have the same people post their same
old "theories" in 2007. How slow is progress...

-Carl

> > Partch's entire theory is in fact founded on the
> > idea that each prime-factor contributes something
> > distinctive to the sound of the music/tuning.
>
> Is there any evidence at all to support this statement?
>
> > According to my theory (and it didn't start with me,
> > cf. Helmholtz, Partch, etc.), the "special" similitude
> > exhibited by the "octave" is simply the affect of
> > prime-factor 2.
>
> In what way did Partch or Helmholtz say... what exactly?

I did a search on google in Helmholtz's "sensation of tone": prime numbers are not named (just a footnote of Ellis about Eulero's theory).

I transcribe from Partch's book "genesis of a music" (page 114):

"The question as to whether the ear prefers a ratio composed of multiples of small prime numbers to a ratio of approximately the same width which involves smaller numbers but a larger prime number is one that cannot be answered categorically; each instance is a different problem. The fairly dose intervals 9/8 (203.9 cents) and 8/7 (231.2 cents) are an example. Considered only as melodic relationships to 1/1, the ear might generally prefer 9/8, in which both numbers are multiples of 3 or less, to 8/7, a ratio with smaller numbers but involving a prime number higher than 3. There are, however, many instances in which a composer will prefer 8/7 for melodic or tonality considerations; the ear by no means always prefers the multiple-number ratio (see below)."

[...]

"To put it in another way, each of these multiple-number ratios is the result of two ratios within the 5 limit:
8/5 X 4/3 = 32/15, which, reduced within a 2/1, is 16/15;
5/3 X 4/3 = 20/9, which, reduced, is 10/9;
3/2 X 3/2 = 9/4, reduced, 9/8;
4/3 X 4/3 = 16/9;
3/2 X 6/5 = 18/10, in reduced terms, 9/5;
and 3/2 X 5/4 = 15/8."

[...]

"In the Pythagorean ratio 81/64 both numbers are multiples of 3 or under, yet because of their excessive largeness the ear certainly prefers 5/4 for this approximate degree, even though it involves a prime number higher than 3.

In the case of the 45/32 "tritone" our theorists have gone around their elbows to reach their thumbs, which could have been reached simply and directly and non-"diabolically" via the number 7. However, since 7 is outside the province of this phase of the discussion, the lacuna 4/3-3/2 will be abeyed, temporarily."

It's interesting that this is written in the chapter "analysis of 5-limit". It seems to me that Partch's thought was more prime-limit directed. The 5 limit includes 15/8, 9/8, 10/9, 16/9, 9/5: it seems a 5-prime limit. Maybe I'm wrong but I think there is nothing in this book about a 9-limit.

For this reason I think Partch was contradicting himself in defining odd-limit ratios families in chapter three:
since Partch's limit goes from 3 to 11, the 9-ratios was the only possibility we have to understand his opinion about it. And it seems to me that in this book prime limit concept prevails in a quite explicit way. Read this (page 150):

"Enigma of the Multiple-Number Ratio

One perplexing problem remains to be touched upon, namely the strength of a multiple-number ratio as compared with a ratio of approximately the same width involving prime numbers or a prime number, the foremost example of which is 9/8 as compared with 8/7. Just why the ear hears 9/8 with speedier conviction than it hears 8/7, either consecutively or simultaneously sounded, can only be conjectured at present. To be sure, 9/8 is the difference between two of the strongest consonances, 3/2 and 4/3, but further, the wave period, from maximum to maximum, has an added regularity in 9/8 - three 3's of the 9 and four 2's of the 8-a frequency of period in air-particle displacement much greater in each of the tones of 9/8. The explanation that suggests itself is that some of this strength is carried over into the simultaneous sounding. With 8/7, it is a clear case of eight waves against seven waves; because of the involvement of the prime number 7, there is no added regularity."

*****

So here is my personal synthesis about odd and prime limit:

Odd-limit

1) it considers only octave similitude instead of other similitudes
2) it fails even with low odd numbers in describing concordance: 8/7 is less concordant than 16/9 (see Partch above)
3) it fails in defining "affects" of ratios: to me 16/9 has a clear "square" flavour as 3/2.
4) as a measure of concordance n*d has to be considered in a different way for odd/even-odd (9/4 - 9/7) or even/odd (16/9 - 10/9)

a) it is not a dense set so it is easier to handle

Prime-limit

1) it's a dense set so it's not easy to handle.
2) it's not clear when p*q is unsustainable and it is necessary to go to the next prime

a) it considers any kind of similitude, not only octaves
b) concordance is well defined within each limit by primes exponents (or n*d)
c) it well defines ratios "affects"

Both

1) both limits are subjected to harmonic entropy: 32/27, 13/11, 19/16 are perceived as a rough 6/5

Just to let you know...I'm a prime-limit fan!

lorenzo

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> > I haven't read more than a few words of Partch, but when his
> > instruments are strummed slowly, it sure sounds like a concrete
> > demonstration of that very statement to me. I could be wrong.
>
> How so?

Literally what I said- like running up the partials: pulling the
drawbars.

> Also, even if it *sounded* that way, it wouldn't establish
> anything about Partch's *theory*.

Baloney. If, for example, doing exactly what I just described as
hearing were a logical result of one interpretation of what Partch's
theory might have been, and another interpretation would not
reasonably produce such a result, which interpretation do you think
I'd tend to credit more weight?

> You mean you've found the 23rd partial is influential in
> shaping timbres,

Certainly, try it yourself.

>or the 23rd prime number is influential in
> the "affect" of dyads?

How many times do I have to say that I'm dubious about "primes"?
Otherwise, that too.

> Everybody's got a theory. Theories must make predictions.
> Exactly what are your theories predicting?

I can't believe it's not clear from what I've been typing hear the
last week or more.

The theory predicts, accurately from what I've heard so far, what
Lorenzo called "undulations" in the rise of dissonance in moving
away from a simple interval. It predicts which "tempered" intervals
are probably going to sound "more like the original". It predicts
which intervals will probably "stick out" in a tuning (when I asked
Lorenzo to tell me which interval "sticks out", it was a downright
rhetorical question) and thereby predict which will better "fit in".
Most importantly to me, though it may be of zero interest to anyone
else, the whole business works in practice for me and is saving me
gobs of time in a specific application.

> Below 23 there are three composite odds, and besides, in
> the context of monz's posts "primes" implies *factoring*.

I'm talking about "character families" of intervals: if I find
distinctive 9, 15 and 21 families, would you advise me to pretend
they don't exist and chase after primes?

> It's a shame that Paul Erlich so patiently showed, over the
> span of a decade on this list, that such claims are
> groundless, only to have the same people post their same
> old "theories" in 2007. How slow is progress...

Which claims do you mean?

-Cameron Bobro

--- In tuning@yahoogroups.com, "Lorenzo Frizzera"
<lorenzo.frizzera@...> wrote:

Thanks, Lorenzo.

Just a couple of comments.
> (Partch) "In the Pythagorean ratio 81/64 both numbers are
multiples of 3 or >under, yet because of their excessive largeness
the ear certainly >prefers 5/4 for this approximate degree, even
though it involves a >prime number higher than 3.

Here's something I've been bitching about since I came here: 81/64
is not an "approximation", it's 81/64. It is born of fifths, not the
fifth partial.

(Lorenzo)
> 1) both limits are subjected to harmonic entropy: 32/27, 13/11,
>19/16 are perceived as a rough 6/5

32/27 is percieved of as 32/27, 13/11 as 13/11, and 19/16 as 19/16.
You have to LEARN to percieve of them as "rough 6/5".

The whole lot would likely be percieved of as simply "out of tune"
by those completely immersed in 12-EDO. (That's not a daring
prediction come to think of it- from personal experience, a very
skilled musician who works in 12-EDO called those very intervals
exactly that. Of course he called 6/5 "out of tune as well...)

-Cameron Bobro

> It's interesting that this is written in the chapter "analysis
> of 5-limit". It seems to me that Partch's thought was more
> prime-limit directed.

Different sections of Genesis use the term "limit" in different
ways. The book, after all, is the story of its author's journey
of discovery. Here, he concludes there is no general rule for
comparing compound ratios to primes, and thus agrees with what
Paul, George, and I are saying.

That's not to say there aren't errors in the book. There are.

None of this mentions "affect". His qualities of emotion,
power, suspense, and so on are assigned to ranges of interval
size within the octave.

>Maybe I'm wrong but I think there is nothing in this book
>about a 9-limit.

Partch puts 9-limit consonances (9/5, etc.) on par with the
other consonances in his 11-limit diamond, but does not include
compounds like 15, 21, or 27.

>"Enigma of the Multiple-Number Ratio
>
>One perplexing problem remains to be touched upon, namely the
>strength of a multiple-number ratio as compared with a ratio of
>approximately the same width involving prime numbers or a prime
>number, the foremost example of which is 9/8 as compared with
>8/7. Just why the ear hears 9/8 with speedier conviction than
>it hears 8/7, either consecutively or simultaneously sounded,
>can only be conjectured at present. To be sure, 9/8 is the
>difference between two of the strongest consonances, 3/2 and
>4/3, but further, the wave period, from maximum to maximum, has
>an added regularity in 9/8 - three 3's of the 9 and four 2's of
>the 8-a frequency of period in air-particle displacement much
>greater in each of the tones of 9/8. The explanation that
>suggests itself is that some of this strength is carried over
>into the simultaneous sounding. With 8/7, it is a clear case
>of eight waves against seven waves; because of the involvement
>of the prime number 7, there is no added regularity."

He's speculating that there's factoring going on, but reaches
no conclusions. On pg 155 he shows the "One Footed Bride"
with 8/7 and 7/4 being stronger consonances than 9/8 and 16/9.

> Odd-limit
//
> 2) it fails even with low odd numbers in describing concordance:
> 8/7 is less concordant than 16/9 (see Partch above)

Partch says 8/7 is less concordant than 9/8 above. He doesn't
mention 16/9.

> 4) as a measure of concordance n*d has to be considered in a
> different way for odd/even-odd (9/4 - 9/7) or even/odd
> (16/9 - 10/9)

How would you adjust its predictions?

> Prime-limit
> b) concordance is well defined within each limit by primes
> exponents (or n*d)

What does this mean?

> c) it well defines ratios "affects"

Again, how can I use it to predict what affects I will hear?

> 1) both limits are subjected to harmonic entropy: 32/27,
> 13/11, 19/16 are perceived as a rough 6/5

This issue is generally called "tolerance".

-Carl

> 32/27 is percieved of as 32/27, 13/11 as 13/11, and 19/16 as 19/16.
> You have to LEARN to percieve of them as "rough 6/5".

I guess this is where we disagree. Assuming that the brain
has a harmonic template of some kind explains some things, though,
like why there are few intervals as consonant as 3:2 near 3:2.

-Carl

> > You mean you've found the 23rd partial is influential in
> > shaping timbres,
>
> Certainly, try it yourself.
//
> > or the 23rd prime number is influential in
> > the "affect" of dyads?
>
> How many times do I have to say that I'm dubious about "primes"?
> Otherwise, that too.

You seemed to be saying that since the 23rd partial is important
to timbre, the 23rd prime should be important to the "affect" of
simultaneous dyads. This may not be completely far-fetched, but
forgive me if I say it certainly doesn't follow without further
clarification.
> > Everybody's got a theory. Theories must make predictions.
> > Exactly what are your theories predicting?
>
> I can't believe it's not clear from what I've been typing hear the
> last week or more.

I'm saying n*d ranks intervals like I hear them, more or less.
Exactly how can I produce the ranking you claim to favor?

> The theory predicts, accurately from what I've heard so far, what
> Lorenzo called "undulations" in the rise of dissonance in moving
> away from a simple interval. It predicts which "tempered" intervals
> are probably going to sound "more like the original". It predicts
> which intervals will probably "stick out" in a tuning (when I asked
> Lorenzo to tell me which interval "sticks out", it was a downright
> rhetorical question) and thereby predict which will better "fit
> in".

I haven't seen anything like this. How can I make these
predictions?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > 32/27 is percieved of as 32/27, 13/11 as 13/11, and 19/16 as 19/16.
> > You have to LEARN to percieve of them as "rough 6/5".
>
> I guess this is where we disagree. Assuming that the brain
> has a harmonic template of some kind explains some things, though,
> like why there are few intervals as consonant as 3:2 near 3:2.
>
> -Carl
>

Perhaps there is a "harmonic template", at this point I'm pretty sure
there is. In fact my little theory assumes a kind of harmonic
template. That doesn't mean that 13/11 is going to heard as a rough
6/5, though. If I've made my idea clear by now, you'll know that I am
up for the idea that we REFERENCE 13/11 to 6/5 as part of our
perception. Their respective characters are so strong, how could
one be heard as a version of the other?

The first few partials are usually very strong, how could we NOT use
them as reference points? But a reference point is not a mold.

-Cameron Bobro

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

> You seemed to be saying that since the 23rd partial is important
> to timbre, the 23rd prime should be important to the "affect" of
> simultaneous dyads. This may not be completely far-fetched, but
> forgive me if I say it certainly doesn't follow without further
> clarification.

Why should I have to forgive you for saying something reasonable?
:-)

First take a listen to "two fifths". Finally got a file up (ran into
some problems on the laptop, arg),I'm trying to get more up ASAP.

http://abumbrislumen.googlepages.com/home

> I'm saying n*d ranks intervals like I hear them, more or less.
> Exactly how can I produce the ranking you claim to favor?

As I keep saying, I'm interested in ranking brute
consonance/dissonance; for one thing, it's so context-dependent.
I'm interesting in understanding at least part of what makes
interval "character", for purely practical purposes.

> I haven't seen anything like this. How can I make these
> predictions?

Well, first listen to the two fifths and we'll see if you think
there might be some merit to the idea that "character families".

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
That should be, NOT interested in ranking brute consonance/dissonance
>
> As I keep saying, I'm interested in ranking brute
> consonance/dissonance; for one thing, it's so context-dependent.

> > You seemed to be saying that since the 23rd partial is important
> > to timbre, the 23rd prime should be important to the "affect" of
> > simultaneous dyads. This may not be completely far-fetched, but
> > forgive me if I say it certainly doesn't follow without further
> > clarification.
>
> Why should I have to forgive you for saying something reasonable?
> :-)
>
> First take a listen to "two fifths". Finally got a file up (ran
> into some problems on the laptop, arg),I'm trying to get more up
> ASAP.
>
> http://abumbrislumen.googlepages.com/home

Um, what am I supposed to be listening for again? Other than
the fact that the second interval beats slightly faster...
There is no difference in consonance or affect here.

> > I'm saying n*d ranks intervals like I hear them, more or less.
> > Exactly how can I produce the ranking you claim to favor?
>
> As I keep saying, I'm interested in ranking brute
> consonance/dissonance; for one thing, it's so context-dependent.
> I'm interesting in understanding at least part of what makes
> interval "character", for purely practical purposes.
>
> > I haven't seen anything like this. How can I make these
> > predictions?
>
> Well, first listen to the two fifths and we'll see if you think
> there might be some merit to the idea that "character families".

I think I'd have to have more than one pair of intervals
to think about families.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> Um, what am I supposed to be listening for again? Other than
> the fact that the second interval beats slightly faster...
> There is no difference in consonance or affect here.

You mean, you don't hear a difference in consonance or affect.

Anyone else?

> I think I'd have to have more than one pair of intervals
> to think about families.

Where are you going to start, by listening to two dozen intervals all
at the same time?

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > Um, what am I supposed to be listening for again? Other than
> > the fact that the second interval beats slightly faster...
> > There is no difference in consonance or affect here.
>
> You mean, you don't hear a difference in consonance or affect.
>
> Anyone else?
>
> > I think I'd have to have more than one pair of intervals
> > to think about families.
>
> Where are you going to start, by listening to two dozen intervals all
> at the same time?
>
> -Cameron Bobro

I'm so glad I'm not here, or I might have been tempted to jump into
this quagmire -- against my better judgment. ;-)

Who started this, anyway?

--George

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>> Um, what am I supposed to be listening for again? Other than
>> the fact that the second interval beats slightly faster...
>> There is no difference in consonance or affect here.
> > You mean, you don't hear a difference in consonance or affect. > > Anyone else?

I can't really detect any difference, but the slow beating of the individual note timbres may be obscuring the difference between the two versions.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> I'm so glad I'm not here, or I might have been tempted to jump into
> this quagmire -- against my better judgment. ;-)

The real quagmire is the soggy bog of assumptions upon which a great
deal of tuning theory seems to be built, as far as I can make out.

> Who started this, anyway?

I thought you did. :-)

-Cameron Bobro

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, "Lorenzo Frizzera" > <lorenzo.frizzera@...> wrote:
>> 1) both limits are subjected to harmonic entropy: 32/27, 13/11, >> 19/16 are perceived as a rough 6/5
> > 32/27 is percieved of as 32/27, 13/11 as 13/11, and 19/16 as 19/16.
> You have to LEARN to percieve of them as "rough 6/5". Maybe so, but you have to learn to perceive any difference between 32/27, 13/11, and 19/16 in the first place. They all sound like slightly flat minor thirds, just flat by different amounts (and it takes a special timbre or harmonic context to perceive any justness about them). Of these, 13/11 is probably the most distinctive, but 32/27 by itself doesn't sound like anything special compared with other intervals in the range (e.g. 51/43). Yes, side by side you can hear a difference between all these intervals in isolation, but in most musical contexts, I doubt very many listeners would notice if you substituted a 51/43 or even a 19/16 for a 32/27 by mistake (unless you have something like a 16/9 in the same chord).

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

>
> I can't really detect any difference, but the slow beating of the
> individual note timbres may be obscuring the difference between the
two
> versions.
>

Okay, looks like the timbre is going to be a problem, so I just
replaced the file with "TwoFifths2", which has a straight sound. :-)

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Maybe so, but you have to learn to perceive any difference between
> 32/27, 13/11, and 19/16 in the first place. They all sound like
>slightly
> flat minor thirds, just flat by different amounts (and it takes a
> special timbre or harmonic context to perceive any justness about
>them).

But they don't sound like flat minor thirds to me, they sound like
different kinds of minor thirds. Different characters.

"My girl is really beautiful. She is just a little shorter than
Sharon Stone, most people wouldn't even notice the difference. Her
hair is almost as blond, about 5.3 percent darker..."

See what I'm driving at?

> Of these, 13/11 is probably the most distinctive,

I agree. These are not things that we necessarily have to think
about- it didn't occur to me until today when I actually looked at
the tuning chart rather than just playing and listening as usual
that I keep migrating in my irregular tunings to the places with
13/11 thirds.

>Yes, side by side you can hear a difference between
> all these intervals in isolation, but in most musical contexts, I
>doubt
> very many listeners would notice if you substituted a 51/43 or
>even a
> 19/16 for a 32/27 by mistake (unless you have something like a
>16/9 in
> the same chord).

I doubt very much if I'd notice. But I try to listen to the whole
first and foremost. A tiny difference in one interval is one thing,
but little differences here and there, and for example a tiny
difference in a generating interval can really change the character
of the whole tuning, I believe.

> The real quagmire is the soggy bog of assumptions upon which a great
> deal of tuning theory seems to be built, as far as I can make out.

It's not that there's nothing going on in your example, it's just
that whatever it is, it's an order of magnitude more subtle than
what's usually discussed in music theory or psychoacoustics.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > The real quagmire is the soggy bog of assumptions upon which a
great
> > deal of tuning theory seems to be built, as far as I can make out.
>
> It's not that there's nothing going on in your example, it's just
> that whatever it is, it's an order of magnitude more subtle than
> what's usually discussed in music theory or psychoacoustics.
>
> -Carl

Maybe the new "straight" version up now is more clear.

Yet as I continue to monkey with these things, every day and a great
deal because they're integral to my music, I find that the effects are
obvious to "laypersons". Sometimes the comments are brilliant- that
one sounds like a synthesizer, that one sounds like an accordian.

How about if I ask, do these sound like perfectly acceptable "fifths"?
No cop-out no-shit-Sherlock answers like "it depends on the context",
just, does it sound like a decent fifth? :-D

If you don't find any difference in consonance and affect, would you
say that they're equally dissonant?

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@> wrote:
>
> >
> > I can't really detect any difference, but the slow beating of the
> > individual note timbres may be obscuring the difference between the
> two
> > versions.
> >
>
> Okay, looks like the timbre is going to be a problem, so I just
> replaced the file with "TwoFifths2", which has a straight sound. :-)
>

... I think I can hear some difference, but can't put a finger on just
what it is. I'd almost wear the second fifth was wider!

Of course it would be much easier if there was no tasteful silence
between the first and the second!

~~~T~~~

----- Original Message -----
From: "Cameron Bobro" <misterbobro@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 08 Ocak 2007 Pazartesi 1:10
Subject: [tuning] Re: Odd limit

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > Do we have free instrument samples already made?
>
> There is a whole bunch of presets, under "banks" IIRC.
>

But if one wishes for more instruments, how can one get them?

>
> > I have a Windows ME platform with a SBLive, Yamaha SW1000XG and DSP
> Factory
> > 2416 audio card. Even the DSP Factory produces latent sounds. I am
> > considering a Macbook pro with all the latest gear in a year or so.
>
> Maybe it's the Windows ME? Because you shouldn't have a problem with
> that setup- can you access the latency settings on your soundcards?
> They're usually set very high by default.
>
>

I noticed that when the program starts in an MS-DOS launch window, there is
a command that reads: "set pa_min_latency_msec=70". But it says also "out of
environment space". Don't know what any of this means. Also, I do not know
how to tweak the latency settings of my hardware.

> Yet as I continue to monkey with these things, every day and a
> great deal because they're integral to my music, I find that the
> effects are obvious to "laypersons".

Things like the fifths demo? Because I've played people music
in 7-limit JI with the tonic wandering all over the place and
they don't notice anything's different.

> How about if I ask, do these sound like perfectly
> acceptable "fifths"?
> No cop-out no-shit-Sherlock answers like "it depends on the
> context", just, does it sound like a decent fifth? :-D

Yes, they were both perfectly acceptable fifths. Can you give
your URL again for the straight version? And was I right that
the second fifth was more out of tune (wider?)?

> If you don't find any difference in consonance and affect, would
> you say that they're equally dissonant?

Not necessarily. I don't think of consonance and dissonance
being strict inverses, though they do have an inverse relationship
throughout much of the first octave.

In this case, though, I would say they are equally dissonant.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Yet as I continue to monkey with these things, every day and a
> > great deal because they're integral to my music, I find that the
> > effects are obvious to "laypersons".
>
> Things like the fifths demo? Because I've played people music
> in 7-limit JI with the tonic wandering all over the place and
> they don't notice anything's different.

Not noticing things like that is a good sign, IMO. Where is it
written in stone that a tonic shouldn't wander? I've played things
in very "alternate" tunings for other musicians and they didn't
even notice, a great compliment. I'm talking about musical effects,
about "I like this one better than that one".

> > How about if I ask, do these sound like perfectly
> > acceptable "fifths"?
> > No cop-out no-shit-Sherlock answers like "it depends on the
> > context", just, does it sound like a decent fifth? :-D
>
> Yes, they were both perfectly acceptable fifths.

I agree. Even if I get nothing else out of all this, I'm tickled
pink that these fifths are clearly fifths to others because...
they're fairly far from 3/2. :-)

>Can you give
> your URL again for the straight version?

Same place as the old one, I replaced it...

http://abumbrislumen.googlepages.com/

>And was I right that
> the second fifth was more out of tune (wider?)?

Yes.

> > If you don't find any difference in consonance and affect, would
> > you say that they're equally dissonant?
>
> Not necessarily. I don't think of consonance and dissonance
> being strict inverses,

that's a good point. I consider dissonance an active, positive
quality, not a lack of consonance.

>though they do have an inverse relationship
> throughout much of the first octave.

Don't understand that.
>
> In this case, though, I would say they are equally dissonant.

I think so too, in spite of... well I'll save that for a bit. One
simply sounds better to me, and as Tom Dent just pointed out to me,
that could be because of judging it first as a melodic interval. In
the end my "character families" may turn out to melodic families,
have to keep that in mind.

> Yes, they were both perfectly acceptable fifths. To me too.
The first is most concordant.
The second has a higher intensity.

lorenzo

>32/27 is percieved of as 32/27, 13/11 as 13/11, and 19/16 as 19/16.
>You have to LEARN to percieve of them as "rough 6/5".
>
>The whole lot would likely be percieved of as simply "out of tune"
>by those completely immersed in 12-EDO. (That's not a daring
>prediction come to think of it- from personal experience, a very
>skilled musician who works in 12-EDO called those very intervals
>exactly that. Of course he called 6/5 "out of tune as well...)

In Scala/edit, just under the "enter value" form, there is a cursor which can be moved with left an right arrows. I'have tried to tune by ear 32/27 and 13/11 but the only ratio that I can tune anytime in the same way is 6/5 (anyway 6/5 is not in 12-EDO).

lorenzo

Hi Carl.

(monz: I really think that the subject here is odd-limit so I changed it again... Very soon I would like to consider "prime affects" and I would return to the "affects" subject!)

>> It's interesting that this is written in the chapter "analysis
>> of 5-limit". It seems to me that Partch's thought was more
>> prime-limit directed.
>
>Different sections of Genesis use the term "limit" in different
>ways.

Do you mean prime and odd-limit?

>The book, after all, is the story of its author's journey
>of discovery. Here, he concludes there is no general rule for
>comparing compound ratios to primes, and thus agrees with what
>Paul, George, and I are saying.

If sometimes he used the term 'limit' in the sense of prime-limit he also agrees with other people.

Page 114. Genesis Of A Music. "...we will keep the scale within the 5 limit for the present. Six new ratios, composed of numbers which are multiples of 5 or under, are added in two of the wide gaps..." These ratios are 16/15 10/9 9/8 16/9 9/5 and 15/8. I would say that here he thought to a 5-prime limit.

>Partch puts 9-limit consonances (9/5, etc.) on par with the
>other consonances in his 11-limit diamond, but does not include
>compounds like 15, 21, or 27.

True. But in the one-footed bride they are included and Partch wrote (page 156) "the fabric involve only ratios based on 11-limit, or small-number multiple-number ratios based on the 11-limit."
This means that 40/27 32/21 27/16 15/8 40/21 64/33 160/81 81/80 33/32 21/20 16/15 32/27 21/16 27/20 are in the 11-limit: prime limit.

Page 125. "Inconsistencies in the 3-limit thinking [...] The Pythagorean scale..." This is a 3-prime limit.

>He's speculating that there's factoring going on, but reaches
>no conclusions. On pg 155 he shows the "One Footed Bride"
>with 8/7 and 7/4 being stronger consonances than 9/8 and 16/9.

On what is based the measure of the length of each ratio?

>> 2) it fails even with low odd numbers in describing concordance:
>> 8/7 is less concordant than 16/9 (see Partch above)
>
>Partch says 8/7 is less concordant than 9/8 above. He doesn't
>mention 16/9.

I was wrong. Anyway he would agree with me that odd limit is not a measure of concordance.

>> 4) as a measure of concordance n*d has to be considered in a
>> different way for odd/even-odd (9/4 - 9/7) or even/odd
>> (16/9 - 10/9)
>
>How would you adjust its predictions?

If I would admit that odd limit is a way to measure concordance (but the 8/7 9/8 example shows me that this is'nt true) for odd/even-odd ratios concordance would be in direct proportion with n*d while for even/odd it would be in inverse proportion.

>> Prime-limit
>> b) concordance is well defined within each limit by primes
>> exponents (or n*d)
>
>What does this mean?

I was wrong.

>> c) it well defines ratios "affects"
>
>Again, how can I use it to predict what affects I will hear?

I will try to answer in another mail.

>> 1) both limits are subjected to harmonic entropy: 32/27,
>> 13/11, 19/16 are perceived as a rough 6/5
>
>This issue is generally called "tolerance".

ok. I have thought that h.e. was also a way to measure tolerance.

lorenzo

> >Different sections of Genesis use the term "limit" in different
> >ways.
>
> Do you mean prime and odd-limit?

Yes, and sometimes it's not clear which meaning is intended.

> >Partch puts 9-limit consonances (9/5, etc.) on par with the
> >other consonances in his 11-limit diamond, but does not include
> >compounds like 15, 21, or 27.
>
> True. But in the one-footed bride they are included and
> Partch wrote (page 156) "the fabric involve only ratios
> based on 11-limit, or small-number multiple-number ratios
> based on the 11-limit."
> This means that 40/27 32/21 27/16 15/8 40/21 64/33 160/81 81/80
> 33/32 21/20 16/15 32/27 21/16 27/20 are in the 11-limit: prime
> limit.

Quite the contrary. If Partch had meant prime limit, why
would he say "certain small-number multiple-number ratios
based on the 11-limit". Clearly here he sees these
ratios as extensions of the 11-odd-limit.

> Page 125. "Inconsistencies in the 3-limit thinking [...] The
> Pythagorean scale..." This is a 3-prime limit.
>
> >He's speculating that there's factoring going on, but reaches
> >no conclusions. On pg 155 he shows the "One Footed Bride"
> >with 8/7 and 7/4 being stronger consonances than 9/8 and 16/9.
>
> On what is based the measure of the length of each ratio?

The graph of the sum of sine waves with frequencies lying in
the various proportions -- a not-necessarily valid approach,
though it may reflect some basic truth about things in the
hearing system.

> >> 2) it fails even with low odd numbers in describing concordance:
> >> 8/7 is less concordant than 16/9 (see Partch above)
> >
> >Partch says 8/7 is less concordant than 9/8 above. He doesn't
> >mention 16/9.
>
> I was wrong. Anyway he would agree with me that odd limit is
> not a measure of concordance.

You seem pretty sure you know the mind of Partch...

> >> 4) as a measure of concordance n*d has to be considered in a
> >> different way for odd/even-odd (9/4 - 9/7) or even/odd
> >> (16/9 - 10/9)
> >
> >How would you adjust its predictions?
>
> If I would admit that odd limit is a way to measure concordance
> (but the 8/7 9/8 example shows me that this is'nt true) for
> odd/even-odd ratios concordance would be in direct proportion
> with n*d while for even/odd it would be in inverse proportion.

I encourage you to come up with a rigorous exposition of your
ideas, including the ranking of intervals it predicts.

> >> 1) both limits are subjected to harmonic entropy: 32/27,
> >> 13/11, 19/16 are perceived as a rough 6/5
> >
> >This issue is generally called "tolerance".
>
> ok. I have thought that h.e. was also a way to measure tolerance.

It could be used that way...

-Carl

> > > Yet as I continue to monkey with these things, every day and a
> > > great deal because they're integral to my music, I find that the
> > > effects are obvious to "laypersons".
> >
> > Things like the fifths demo? Because I've played people music
> > in 7-limit JI with the tonic wandering all over the place and
> > they don't notice anything's different.
>
> Not noticing things like that is a good sign, IMO. Where is it
> written in stone that a tonic shouldn't wander? I've played things
> in very "alternate" tunings for other musicians and they didn't
> even notice, a great compliment. I'm talking about musical effects,
> about "I like this one better than that one".
>
> > > How about if I ask, do these sound like perfectly
> > > acceptable "fifths"?
> > > No cop-out no-shit-Sherlock answers like "it depends on the
> > > context", just, does it sound like a decent fifth? :-D
> >
> > Yes, they were both perfectly acceptable fifths.
>
> I agree. Even if I get nothing else out of all this, I'm tickled
> pink that these fifths are clearly fifths to others because...
> they're fairly far from 3/2. :-)

Heh. 3:2 enjoys a huge basin of attraction. Does 720 cents
sound like a fifth to you?

> >Can you give
> > your URL again for the straight version?
>
> Same place as the old one, I replaced it...
>
> http://abumbrislumen.googlepages.com/

Just trying to impress upon posters the importance of
reiterating urls for their readers, some of whom didn't
bookmark your site or have time to search back through
the thread.

The difference is much more pronounced this time.
The latter one sounds worse both in that it beats
faster and in that it has some "chorus" to it, which
I hate.

> > And was I right that
> > the second fifth was more out of tune (wider?)?
>
> Yes.

Woohoo!

> > > If you don't find any difference in consonance and affect, would
> > > you say that they're equally dissonant?
> >
> > Not necessarily. I don't think of consonance and dissonance
> > being strict inverses,
//
> >though they do have an inverse relationship
> > throughout much of the first octave.
>
> Don't understand that.

Between 1/1 and 2/1, they mostly obey an inverse relation.
Beyond 2/1, they both start to drop off, and by the time you
get you 6/1 there really isn't a lot of interaction either
way.

-Carl

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> Okay, looks like the timbre is going to be a problem, so I just
> replaced the file with "TwoFifths2", which has a straight sound. :-)

Good. The version I had not only went waa-waa, it seemed to be corrupt.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> ... I think I can hear some difference, but can't put a finger on just
> what it is. I'd almost wear the second fifth was wider!

So far I've not seen a url.

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

> Heh. 3:2 enjoys a huge basin of attraction.

It's a lumpy region, as far as I can tell.

"The third partial, whether or not it lies at an exact integer
relationship to the fundamental, is very strong in many timbres, and
how other timbres interact with it is very important " is how I'd
put it.

> Does 720 cents
> sound like a fifth to you?

I haven't come across anything at 720, give or take a fraction of a
cent, that has the character, but I did come aross one at 722+ cents
that did. I can't find the exact ratio, but I seem to remember that
very "fifth" being mentioned once somewhere around here, perhaps
I'll find it in Gene's list of high fifths.

> The difference is much more pronounced this time.
> The latter one sounds worse both in that it beats
> faster and in that it has some "chorus" to it, which
> I hate.

Okay!

> (About consonance and dissonance...)

> Between 1/1 and 2/1, they mostly obey an inverse relation.
> Beyond 2/1, they both start to drop off, and by the time you
> get you 6/1 there really isn't a lot of interaction either
> way.

Well, I won't take your word for it. :-)

-Cameron Bobro

Hi Cameron,

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > and
> > I think it's completely independent of the concordance
> > and discordance question
>
> I agree, which is why I keep harping on about character families
> rather than brute consonance/dissonance, which is so very context-
> dependent.

Note that i was careful to use concordance/discordance
instead of consonance/dissonance, and Carl follows my
usage. If you look up "accordance" and "sonance" in
my Encyclopedia, you'll see that several years ago
many of us decided on a convention to use "accordance"
to describe situations devoid of any musical context,
and to restrict the use of "sonance" to situations
dependant on musical context.

-monz
http://tonalsoft.com
Tonescape microtonal music software

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

> The difference is much more pronounced this time.
> The latter one sounds worse both in that it beats
> faster and in that it has some "chorus" to it, which
> I hate.

I'm still not downloading correctly, but from what I got of the second
one I agree; the first seems more concordant.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Cameron,
>
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
>
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > >
> > > and
> > > I think it's completely independent of the concordance
> > > and discordance question
> >
> > I agree, which is why I keep harping on about character families
> > rather than brute consonance/dissonance, which is so very
context-
> > dependent.
>
>
> Note that i was careful to use concordance/discordance
> instead of consonance/dissonance, and Carl follows my
> usage. If you look up "accordance" and "sonance" in
> my Encyclopedia, you'll see that several years ago
> many of us decided on a convention to use "accordance"
> to describe situations devoid of any musical context,
> and to restrict the use of "sonance" to situations
> dependant on musical context.

Seems like a good convention. I try to follow some of the
conventions here because I think they're valuable contributions,
like "EDO" instead of "ET", because you can divide any interval
equally, and "ET" refers specifically to tempering- in many equal
divisions, noone is "tempering" anything, they're simply dividing
evenly.

In this case I'll follow the convention. But until I discover
for the first time in my life something outside of a musical
context, I'll continuing using "-ance".

-Cameron Bobro

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> I haven't come across anything at 720, give or take a fraction of a
> cent, that has the character, but I did come aross one at 722+ cents
> that did. I can't find the exact ratio, but I seem to remember that
> very "fifth" being mentioned once somewhere around here, perhaps
> I'll find it in Gene's list of high fifths.

Heh. 38/25? 44/29??

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
>
> > I haven't come across anything at 720, give or take a fraction
of a
> > cent, that has the character, but I did come aross one at 722+
cents
> > that did. I can't find the exact ratio, but I seem to remember
that
> > very "fifth" being mentioned once somewhere around here, perhaps
> > I'll find it in Gene's list of high fifths.
>
> Heh. 38/25? 44/29??
>

38/25 has a strong "thirdsy" and minor feeling to it, not "fifth" in
character at all. 44/29 sounds kind of neutered but definitely a
fifth.

Okay, I'll apply my idea to find a fifth at 720 that is strongly
fifth in character... 97/64 at 719.895. Yes that sounds very "fifth"
to me, try it.

-Cameron Bobro

>> True. But in the one-footed bride they are included and
>> Partch wrote (page 156) "the fabric involve only ratios
>> based on 11-limit, or small-number multiple-number ratios
>> based on the 11-limit."
>> This means that 40/27 32/21 27/16 15/8 40/21 64/33 160/81 81/80
>> 33/32 21/20 16/15 32/27 21/16 27/20 are in the 11-limit: prime
>> limit.
>
>Quite the contrary. If Partch had meant prime limit, why
>would he say "certain small-number multiple-number ratios
>based on the 11-limit". Clearly here he sees these
>ratios as extensions of the 11-odd-limit.

In which way 160/81 is an extension of 11-odd limit?

>> >He's speculating that there's factoring going on, but reaches
>> >no conclusions. On pg 155 he shows the "One Footed Bride"
>> >with 8/7 and 7/4 being stronger consonances than 9/8 and 16/9.
>>
>> On what is based the measure of the length of each ratio?
>
>The graph of the sum of sine waves with frequencies lying in
>the various proportions

I don't understand. Can you do an example?

>> >Partch says 8/7 is less concordant than 9/8 above. He doesn't
>> >mention 16/9.
>>
>> I was wrong. Anyway he would agree with me that odd limit is
>> not a measure of concordance.
>
>You seem pretty sure you know the mind of Partch...

I'm not a medium :-) I have just read this (page 114):

"The fairly dose intervals 9/8 (203.9 cents) and 8/7 (231.2 cents) are an example. Considered only as melodic relationships to 1/1, the ear might generally prefer 9/8..."

"Just why the ear hears 9/8 with speedier conviction than it hears 8/7, either consecutively or simultaneously sounded, can only be conjectured at present."

I still think he would agree with me that odd limit is not a measure of concordance.

>> >> 4) as a measure of concordance n*d has to be considered in a
>> >> different way for odd/even-odd (9/4 - 9/7) or even/odd
>> >> (16/9 - 10/9)
>> >
>> >How would you adjust its predictions?
>>
>> If I would admit that odd limit is a way to measure concordance
>> (but the 8/7 9/8 example shows me that this is'nt true) for
>> odd/even-odd ratios concordance would be in direct proportion
>> with n*d while for even/odd it would be in inverse proportion.
>
>I encourage you to come up with a rigorous exposition of your
>ideas, including the ranking of intervals it predicts.

At the moment I really can't. Infact n*d, in odd or prime limit, has no significance in itself because it works only with small numbers. 307/306 is an odd and prime limit. The density of partials (which considering time is the vertical density of sound) and the neural firing (horizontal density), which can be both measured by n*d, is high. But it sounds quite good. Why? I can't answer. Without this answer it will not be possible to use n*d as a concordance measure. Do you have some ideas?

lorenzo

> > I agree, which is why I keep harping on about character families
> > rather than brute consonance/dissonance, which is so very context-
> > dependent.
>
> Note that i was careful to use concordance/discordance
> instead of consonance/dissonance, and Carl follows my
> usage.

I used to, but in this thread I've kinda given up. I should
go back to it, though.

> If you look up "accordance" and "sonance" in
> my Encyclopedia, you'll see that several years ago
> many of us decided on a convention to use "accordance"

I've never heard "accordance" before. Who agreed to that?

> to describe situations devoid of any musical context,
> and to restrict the use of "sonance" to situations
> dependant on musical context.

You're the only one I've ever seen use "sonance", and if I
may say so, your blurring of the line between an encyclopedia
and a exposition of your own ideas was rather unprofessional.

-Carl

> >> True. But in the one-footed bride they are included and
> >> Partch wrote (page 156) "the fabric involve only ratios
> >> based on 11-limit, or small-number multiple-number ratios
> >> based on the 11-limit."
> >> This means that 40/27 32/21 27/16 15/8 40/21 64/33 160/81 81/80
> >> 33/32 21/20 16/15 32/27 21/16 27/20 are in the 11-limit: prime
> >> limit.
> >
> >Quite the contrary. If Partch had meant prime limit, why
> >would he say "certain small-number multiple-number ratios
> >based on the 11-limit". Clearly here he sees these
> >ratios as extensions of the 11-odd-limit.
>
> In which way 160/81 is an extension of 11-odd limit?

Lorenzo, if Partch had meant 11-prime-limit, why would he
say '11-limit plus some other multiple-number ratios'?

> >> >He's speculating that there's factoring going on, but reaches
> >> >no conclusions. On pg 155 he shows the "One Footed Bride"
> >> >with 8/7 and 7/4 being stronger consonances than 9/8 and 16/9.
> >>
> >> On what is based the measure of the length of each ratio?
> >
> >The graph of the sum of sine waves with frequencies lying in
> >the various proportions
>
> I don't understand. Can you do an example?

See the photographs in Partch's book near pg. 155.

> >> >Partch says 8/7 is less concordant than 9/8 above. He doesn't
> >> >mention 16/9.
> >>
> >> I was wrong. Anyway he would agree with me that odd limit is
> >> not a measure of concordance.
> >
> >You seem pretty sure you know the mind of Partch...
>
> I'm not a medium :-) I have just read this (page 114):
>
> "The fairly dose intervals 9/8 (203.9 cents) and 8/7
> (231.2 cents) are an example. Considered only as melodic
> relationships to 1/1, the ear might
> generally prefer 9/8..."

You said "concordance", but Partch is talking about melodic
intervals here. And further, Partch does not state that odd
limit is not a measure of something.

> >I encourage you to come up with a rigorous exposition of your
> >ideas, including the ranking of intervals it predicts.
>
> At the moment I really can't. Infact n*d, in odd or prime limit,
> has no significance in itself because it works only with small
> numbers. 307/306 is an odd and prime limit. The density of
> partials (which considering time is the vertical density of
> sound) and the neural firing (horizontal density), which can
> be both measured by n*d, is high. But it sounds quite good. Why?
> I can't answer. Without this answer it will not be possible to
> use n*d as a concordance measure. Do you have some ideas?

It's my belief that n*d ranks ratios between 1/1 and 4/1
by increasing discordance better than any other simple
method, when the product n*d < 60. But I'm willing to
change that belief in light of evidence to the contrary.

To evaluate ratios with n*d > 60, I use harmonic entropy, and
I think it's the simplest method that ranks intervals regardless
of the size of numbers in the ratios. Again, I'm willing
to change that belief.

-Carl

I sent this yesterday but Yahoo bounced it. Stupid Yahoo. I hate it.

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>>> The real quagmire is the soggy bog of assumptions upon which a > great >>> deal of tuning theory seems to be built, as far as I can make out. >> It's not that there's nothing going on in your example, it's just
>> that whatever it is, it's an order of magnitude more subtle than
>> what's usually discussed in music theory or psychoacoustics.
>>
>> -Carl
> > Maybe the new "straight" version up now is more clear. > > Yet as I continue to monkey with these things, every day and a great > deal because they're integral to my music, I find that the effects are > obvious to "laypersons". Sometimes the comments are brilliant- that > one sounds like a synthesizer, that one sounds like an accordian.

Both of the ones in the previous version sounded like a synthesizer,
with a specific kind of slow timbral shifting (almost a flanging effect)
that I've heard in recordings of slightly detuned synthesizers. The
current "straight" versions sound more natural, and make the difference
more apparent.

> How about if I ask, do these sound like perfectly acceptable "fifths"?
> No cop-out no-shit-Sherlock answers like "it depends on the context",
> just, does it sound like a decent fifth? :-D
> > If you don't find any difference in consonance and affect, would you > say that they're equally dissonant?
> > -Cameron Bobro

Yep, they're perfectly reasonable fifths. Don't ask me, though; I write
music using way detuned fifths all the time. :-)

Now if I heard one of these in a piece that was otherwise in miracle or
some other low-error temperament, I might suspect a wrong note (perhaps
deliberate?), but unless the whole thing is in straight JI, there's
nothing especially jarring about either of them. I really don't know
which one I prefer. The second one actually seems a bit more pleasant,
even though it's pretty far from just. Any relation to the golden ratio,
by any chance?

I sent this yesterday but Yahoo bounced it. Stupid Yahoo. I hate it.

Cameron Bobro wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> > >> Maybe so, but you have to learn to perceive any difference between >> 32/27, 13/11, and 19/16 in the first place. They all sound like >> slightly >> flat minor thirds, just flat by different amounts (and it takes a >> special timbre or harmonic context to perceive any justness about >> them).
> > But they don't sound like flat minor thirds to me, they sound like > different kinds of minor thirds. Different characters. Well, I guess "flat" doesn't really give the right impression of how
these intervals sound. The point is that recognizing any difference at
all between 32/27 and 19/16 (which is implicit in the idea that 32/27 is
perceived as specifically 32/27 and not a general "minor third") is an
especially subtle distinction. I don't doubt that trained ears can hear
the difference, but it takes as much "learning" as perceiving these as a
"rough 6/5" would. These intervals have a particular sort of sound,
which I was just calling "flat minor thirds" for lack of a better word
(although 13/11 is on the edge of being almost a subminor third, having
a bit of the flavor of both categories).

>> Yes, side by side you can hear a difference between >> all these intervals in isolation, but in most musical contexts, I >> doubt >> very many listeners would notice if you substituted a 51/43 or >> even a >> 19/16 for a 32/27 by mistake (unless you have something like a >> 16/9 in >> the same chord).
> > I doubt very much if I'd notice. But I try to listen to the whole > first and foremost. A tiny difference in one interval is one thing, > but little differences here and there, and for example a tiny > difference in a generating interval can really change the character > of the whole tuning, I believe. That much is true; small differences add up. Certainly in the context of
a more elaborate tuning system, you can have circumstances in which a
difference of a few cents (513/512 is a little over 3 cents), or even a
few fractions of a cent, can change the character of the system as a whole.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> The point is that recognizing any difference at
> all between 32/27 and 19/16 (which is implicit in the idea that
>32/27 is
> perceived as specifically 32/27 and not a general "minor third")
>is an
> especially subtle distinction.

Yes. I don't don't think that literally recognizing is very
important though. A person doesn't have know the difference between
say fennel and aniseed, or even be able to distinguish them in a
blind test, in order to say "This curry tastes better to me than
that one".

>I don't doubt that trained ears can hear
> the difference, but it takes as much "learning" as perceiving
>these as a
> "rough 6/5" would.

Probably. But if there's just a feeling that they're a little
different in character, even if a person couldn't even tell which
one is higher, for example, that's enough when we're talking about
making tunings ans "temperaments", see your own comment below.

>These intervals have a particular sort of sound,
> which I was just calling "flat minor thirds" for lack of a better
>word

Yes I knew what you meant of course, I'm just belaboring a point.
:-)

> (although 13/11 is on the edge of being almost a subminor third,
>having
> a bit of the flavor of both categories).

Completely agree- I find it a real sweatheart in practice.

> That much is true; small differences add up. Certainly in the
>context of
> a more elaborate tuning system, you can have circumstances in
>which a
> difference of a few cents (513/512 is a little over 3 cents), or
>even a
> few fractions of a cent, can change the character of the system as
>a whole.

Yes. I believe that not just the fact that things add up using a
generating interval, but the very character of the generating is
important.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Both of the ones in the previous version sounded like a
>synthesizer,
> with a specific kind of slow timbral shifting (almost a flanging
>effect)
> that I've heard in recordings of slightly detuned synthesizers. The
> current "straight" versions sound more natural, and make the
>difference
> more apparent.

The first version used mild PWM, so the spectral content is
shifting, a similar color effect to detuning. Heavy PWM actually
does detune so I avoided it.

>
> Yep, they're perfectly reasonable fifths. Don't ask me, though; I
>write
> music using way detuned fifths all the time. :-)
>
> Now if I heard one of these in a piece that was otherwise in
>miracle or
> some other low-error temperament, I might suspect a wrong note
(perhaps
> deliberate?), but unless the whole thing is in straight JI, there's
> nothing especially jarring about either of them. I really don't
know
> which one I prefer. The second one actually seems a bit more
pleasant,
> even though it's pretty far from just. Any relation to the golden
ratio,
> by any chance?

I also find the second one more pleasant- the first one sounds sour
to me. Like it says "tempered" rather than just being itself.

The fifths are 414/275 at 708.239 and 529/351 at 710.156.
Odd number, harmonic entropy and n*d all account for the previous
responses, but they don't account for yours or mine.

I don't know why the flavors seem different but my working theory
(working as in actual practice, and it seems to work) is to take into
consideration the denominator, and octave reduce it, getting a feel
for "character families". Ratios on the overtones "refer to
themselves" so to speak- how could the 720 cent fifth of 97/64 not
sound good?

The two examples here are funky ones, landing at about 123 and 546
cents respectively. What those mean, I don't know yet, need more
examples. But I know that if I have some intervals, and when reduced
in this way only one falls into a strange character family, it's
probably going to be the stinker.

Of course I'm talking about this backwards- I found the stinkers by
ear, then went in to find out why, then apply the idea actually
creating non-stinkers.

What makes this so important to me as that I'm doing something funky
as far as tunings and harmony, and any characteristics pile up
exponentially ( writing it up, but slowly because my time is spent
actually making music with the idea). An acceptable but slightly
smelly interval in a scale is going to breed exponentially and put a
heavy stench in the whole garden in what I'm doing- this must be
true to an extent with any generating interval and I'm sure this
must be very true for those working with "diamonds" and such as well.

take care and thanks for responses guys,

-Cameron Bobro

> The fifths are 414/275 at 708.239 and 529/351 at 710.156.
> Odd number, harmonic entropy and n*d all account for the previous
> responses, but they don't account for yours or mine.

You can't use odd limit or n*d on numbers this big.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > The fifths are 414/275 at 708.239 and 529/351 at 710.156.
> > Odd number, harmonic entropy and n*d all account for the previous
> > responses, but they don't account for yours or mine.
>
> You can't use odd limit or n*d on numbers this big.
>
> -Carl

How big can the numbers be?

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> The fifths are 414/275 at 708.239 and 529/351 at 710.156.
> Odd number, harmonic entropy and n*d all account for the previous
> responses, but they don't account for yours or mine.

None of those, I suspect, explain what is really going on. You need to
compare these to things like the 22-et fifth at 709.0909... cents, it
seems to me, and see if these rational numbers even matter.

> > > The fifths are 414/275 at 708.239 and 529/351 at 710.156.
> > > Odd number, harmonic entropy and n*d all account for the previous
> > > responses, but they don't account for yours or mine.
> >
> > You can't use odd limit or n*d on numbers this big.
>
> How big can the numbers be?

Odd limit works up to somewhere between 7 and 17 depending
on who you ask, and n*d works until the product is around 40-60.

-Carl

Cameron Bobro wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> >
> > > The fifths are 414/275 at 708.239 and 529/351 at 710.156.
> > > Odd number, harmonic entropy and n*d all account for the
previous
> > > responses, but they don't account for yours or mine.
> >
> > You can't use odd limit or n*d on numbers this big.
> >
> > -Carl
>
> How big can the numbers be?

I've got another theory. First, what is your *personal* tolerance
for error in tuning: 2 cents? half a cent? one-third of a cent?

Then, within the range of each of these intervals plus or minus your
tolerance, do you find simpler ratios that you might be assimilating
them to?

Or in other words, does either interval lie in a basin of attraction
of a simpler ratio?

As a measure of ratio simplicity, log (n*d) probably works pretty
well.

For example, can we approximate 529/351 by, say 528/350 = 254/175?
I haven't yet checked how many cents in either; but supposing we
could, does 254/175 lie within your tolerance of 529/351? If so, you
might mentally be assimilating the more complex ratio to the simpler
one. Then 254/175 being a simpler ratio than 414/275 on the log(n*d)
measure, means you could just use n*d.

This might go some way to answering the question you asked:
> How big can the numbers be?

The answer I'm proposing is "no bigger than your *personal* error
tolerance in tuning allows".

The only way to see whether this works is to plug in some numbers and
test. But the answer will be different for different listeners.

Regards,
Yahya

Hi Yahya,

> For example, can we approximate 529/351 by, say 528/350 = 254/175?
> I haven't yet checked how many cents in either; but supposing we
> could, does 254/175 lie within your tolerance of 529/351? If so,
> you might mentally be assimilating the more complex ratio to the
> simpler one. Then 254/175 being a simpler ratio than 414/275 on
> the log(n*d) measure, means you could just use n*d.

I get your drift, but in this case I don't think anyone can hear
numbers this big without lots of highly specialized training. I
know it might have only been a convenient example.

The other point I'd like to make is that a fixed tolerance might
not be the right thing. One key observation of Partch is that the
simpler ratios have a much stronger "pull" than the complex ones.
So the tolerance band that includes 3:2 should be larger than the
one that includes 254:175, and if they're in the same band the band
should be called the 3:2 band.

> > This might go some way to answering the question you asked:
> > How big can the numbers be?
>
> The answer I'm proposing is "no bigger than your *personal* error
> tolerance in tuning allows".

Harmonic entropy actually has a parameter similar to tolerance,
called "s". The smaller s is, the narrower all the bands are, but
they still aren't equal in width. S was found to vary between
listeners in psychoacoustics experiments.

-Carl

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> Cameron Bobro wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > >
> > > > The fifths are 414/275 at 708.239 and 529/351 at 710.156.
> > > > Odd number, harmonic entropy and n*d all account for the
> previous
> > > > responses, but they don't account for yours or mine.
> > >
> > > You can't use odd limit or n*d on numbers this big.
> > >
> > > -Carl
> >
> > How big can the numbers be?
>
> I've got another theory. First, what is your *personal* tolerance
> for error in tuning: 2 cents? half a cent? one-third of a cent?

One of the points I'm trying to make is that "tolerance", mine at
least, is based on character, not on rank physical dimension. The
two intervals in this example are a tiny bit less than 2 cents
apart, but sound different. For all I know, there's an interval 10
cents away that shares the same character as the second interval
here, for example. And character is shared amongst completely
different intervals- you can have a fifth with a somehow "thirds"
character, for example.

> Then, within the range of each of these intervals plus or minus
>your
> tolerance, do you find simpler ratios that you might be
>assimilating
> them to?

I think I assimilate to character families.
>
> Or in other words, does either interval lie in a basin of
>attraction
> of a simpler ratio?

Thinking about it- I don't buy into "basin of attraction".

> As a measure of ratio simplicity, log (n*d) probably works pretty
> well.
>
> For example, can we approximate 529/351 by, say 528/350 = 254/175?
> I haven't yet checked how many cents in either; but supposing we
> could, does 254/175 lie within your tolerance of 529/351? If so,
>you
> might mentally be assimilating the more complex ratio to the
>simpler
> one. Then 254/175 being a simpler ratio than 414/275 on the log>
(n*d)
> measure, means you could just use n*d.

That could very well be, it makes sense. In this case it doesn't
test my theory because the denominators octave-reduce only three
cents apart.
>
> This might go some way to answering the question you asked:
> > How big can the numbers be?
>
> The answer I'm proposing is "no bigger than your *personal* error
> tolerance in tuning allows".
>
> The only way to see whether this works is to plug in some numbers
and
> test. But the answer will be different for different listeners.

Thanks Yayha! this a good idea, and compatible with the sensation of
hearing character. At some point what you describe must be going on,
because recently I've been dealing with some very big ratios and
there's simply no way I'm hearing them literally. When I write up my
hopefully entertaining approach to harmonies, you'll see why I keep
poking at this crap.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Hi Yahya,
>
> > For example, can we approximate 529/351 by, say 528/350 =
254/175?
> > I haven't yet checked how many cents in either; but supposing we
> > could, does 254/175 lie within your tolerance of 529/351? If so,
> > you might mentally be assimilating the more complex ratio to the
> > simpler one. Then 254/175 being a simpler ratio than 414/275 on
> > the log(n*d) measure, means you could just use n*d.
>
> I get your drift, but in this case I don't think anyone can hear
> numbers this big without lots of highly specialized training. I
> know it might have only been a convenient example.

It seems to me that Yahya is saying that we're NOT hearing numbers
this big, but reducing them, and I agree. To what we're reducing is
the question- I think the "nearest simple interval" is bunk,
otherwise it seems unlikely that I'm hearing 710 cents as
more "fifth" than 708.

> The other point I'd like to make is that a fixed tolerance might
> not be the right thing.

I agree, see my post to Yahya.

>One key observation of Partch is that the
> simpler ratios have a much stronger "pull" than the complex ones.

I feel the pull of character, not specific intervals. Simple
intervals are often repellant: reference points for sure, but
demanding beating to come alive. And when I listen to music from
around the world, I'm sure I'm not the only one who hears this way.

-Cameron Bobro

> > I get your drift, but in this case I don't think anyone can hear
> > numbers this big without lots of highly specialized training. I
> > know it might have only been a convenient example.
>
> It seems to me that Yahya is saying that we're NOT hearing numbers
> this big, but reducing them,

If there are no simpler ratios in the band, Yahya's proposal
is that we'll hear the original ratio.

-Carl

Hi Carl,

Carl Lumma wrote:
>
> Hi Yahya,
>
> > For example, can we approximate 529/351 by, say 528/350 = 254/175?
> > I haven't yet checked how many cents in either; but supposing we
> > could, does 254/175 lie within your tolerance of 529/351? If so,
> > you might mentally be assimilating the more complex ratio to the
> > simpler one. Then 254/175 being a simpler ratio than 414/275 on
> > the log(n*d) measure, means you could just use n*d.
>
> I get your drift, but in this case I don't think anyone can hear
> numbers this big without lots of highly specialized training. I
> know it might have only been a convenient example.

Yes, to both points.

> The other point I'd like to make is that a fixed tolerance might
> not be the right thing. One key observation of Partch is that the
> simpler ratios have a much stronger "pull" than the complex ones.
> So the tolerance band that includes 3:2 should be larger than the
> one that includes 254:175, and if they're in the same band the band
> should be called the 3:2 band.

Yes, that makes sense. But could still be accommodated in a theory
of the type I outlined. Do we have any experimental evidence of how
the tolerances vary with interval? By "experimental evidence", of
course, I mean in a properly designed and conducted scientific
experiment, not just anecdotal evidence.

> > > This might go some way to answering the question you asked:
> > > How big can the numbers be?
> >
> > The answer I'm proposing is "no bigger than your *personal* error
> > tolerance in tuning allows".
>
> Harmonic entropy actually has a parameter similar to tolerance,
> called "s". ...

Yes, I noticed that, and you may recall, asked how to determine a
partcular listener's value of s - without any really satisfactory
answer that I remember.

> ... The smaller s is, the narrower all the bands are, but
> they still aren't equal in width. ...

Do they vary with interval in exactly the same way as the listener's
tolerance? Is there even any experimental evidence that tells us how
they vary with interval?

> ... S was found to vary between
> listeners in psychoacoustics experiments.

Where might I find reports of these experiments?

> -Carl

Regards,
Yahya

Hi Cameron,

Cameron Bobro wrote:
>
> --- In tuning@yahoogroups.com, "yahya_melb" <yahya@> wrote:
> >
> > Cameron Bobro wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > > >
> > > > > The fifths are 414/275 at 708.239 and 529/351 at 710.156.
> > > > > Odd number, harmonic entropy and n*d all account for the
> > previous
> > > > > responses, but they don't account for yours or mine.
> > > >
> > > > You can't use odd limit or n*d on numbers this big.
> > > >
> > > > -Carl
> > >
> > > How big can the numbers be?
> >
> > I've got another theory. First, what is your *personal*
tolerance for error in tuning: 2 cents? half a cent? one-third of a
cent?
>
> One of the points I'm trying to make is that "tolerance", mine at
> least, is based on character, not on rank physical dimension. The
> two intervals in this example are a tiny bit less than 2 cents
> apart, but sound different. ...

I understand this point, but if you can discriminate between
intervals less than 2 cents apart, your tolerance for error in tuning
is clearly less than 2 cents. Should I define what I mean by
tolerance? (It doesn't invalidate near intervals having different
characters.)

Definition 1: _A listener's tolerance for error in tuning a
particular interval_ is the largest amount that interval can be
changed by, yet still seem to that listener to be the same interval.

Definition 2: _A listener's tolerance for error in tuning_ is the
smallest value of that listener's tolerance for error in tuning an
interval, taken over all intervals that listener can hear.

Assumption: All intervals are played in a standard natural tuned
acoustic timbre with which the listener is familiar.

> ... For all I know, there's an interval 10
> cents away that shares the same character as the second interval
> here, for example. And character is shared amongst completely
> different intervals- you can have a fifth with a somehow "thirds"
> character, for example.

Yep, I'm content that that ("character") describes something that
some listeners hear.

> > Then, within the range of each of these intervals plus or minus
your tolerance, do you find simpler ratios that you might be
assimilating them to?
>
> I think I assimilate to character families.

I guess I should have written:
"Within the range of each of these intervals plus or minus that
listener's tolerance for error, are there objectively simpler
ratios? If so, that listener might be assimilating them to those
ratios."

> > Or in other words, does either interval lie in a basin of
> > attraction of a simpler ratio?
>
> Thinking about it- I don't buy into "basin of attraction".

Perhaps that means: It doesn't describe what you hear? Tell me, does
the "fifthness" of intervals very near to 3/2 not tend to swamp any
other character they may express? For example, when you hear a 73/49
or a 74/49, does it sound like a fifth, or does it sound distinctly
septimal? You know, I think it might be a valuable experiment for
you to generate a large random set of JI intervals in, say, the 99-
odd-limit; then listen to them presented randomly (not more than
about 20 at a sitting, to avoid fatigue effects) and assign each to
one of a predefined list of characters, eg,
unison-like,
octave-like,
fifth-like,
fourth-like,
major-third-like,
minor-sixth-like,
minor-third-like,
major-sixth-like,
...
major-semitone-like,
minor-semitone-like,
...
Pyth.-comma-like,
...
NONE.

(Your list of characters might be quite different; this is just a
suggested starting point.)

Then analyse the results to see whether the experiment supports your
feeling that you are hearing these kinds of character; in particular,
which characters you strongly (95% of the time or better) associate
to ratios involving a particular prime in the numerator; and
separately, which you strongly associate to ratios involving a
particular prime in the denominator.

There'd be no point me doing such an experiment, I think, because I'm
not at all sure I hear such character families. But you, and others
who do, would be suitable experimental subjects. And the effort
might shed new light on what is going on in perception of musical
intervals - that's one reason I stipulated the use of only natural
tuned acoustic instruments, so the result might be relevant to music.

> > As a measure of ratio simplicity, log (n*d) probably works pretty
> > well.
> >
> > For example, can we approximate 529/351 by, say 528/350 = 254/175?
> > I haven't yet checked how many cents in either; but supposing we
> > could, does 254/175 lie within your tolerance of 529/351? If so,
> >you
> > might mentally be assimilating the more complex ratio to the
> >simpler
> > one. Then 254/175 being a simpler ratio than 414/275 on the log>
> (n*d)
> > measure, means you could just use n*d.
>
> That could very well be, it makes sense. In this case it doesn't
> test my theory because the denominators octave-reduce only three
> cents apart.

What does that mean? I'd hazard a guess, but I doubt you could
possibly mean what I think you mean here ... Please show me the
numbers. (Or, the money would be fine. Really.)

> > This might go some way to answering the question you asked:
> > > How big can the numbers be?
> >
> > The answer I'm proposing is "no bigger than your *personal* error
tolerance in tuning allows".
> >
> > The only way to see whether this works is to plug in some numbers
and test. But the answer will be different for different listeners.
>
> Thanks Yayha! this a good idea, and compatible with the sensation
of hearing character. At some point what you describe must be going
on, because recently I've been dealing with some very big ratios and
there's simply no way I'm hearing them literally. When I write up my
hopefully entertaining approach to harmonies, you'll see why I keep
poking at this crap.

Not crap at all, Cameron; if anyone understood this stuff better than
we yet do, you'd think they'd have explained it by now in a fairly
publicly-accessible way. Personally, I think we've a very long way
to go before we can effectively *describe* the varieties of human
listening experience - much less *explain* them.

Regards,
Yahya

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
> Hi Cameron,

> >
> > One of the points I'm trying to make is that "tolerance", mine
>>at
> > least, is based on character, not on rank physical dimension.
>>The
> > two intervals in this example are a tiny bit less than 2 cents
> > apart, but sound different. ...
>
> I understand this point, but if you can discriminate between
> intervals less than 2 cents apart, your tolerance for error in
>tuning
> is clearly less than 2 cents.

I understand, but would never claim to hear such a small difference.

>Should I define what I mean by
> tolerance? (It doesn't invalidate near intervals having different
> characters.)
>
> Definition 1: _A listener's tolerance for error in tuning a
> particular interval_ is the largest amount that interval can be
> changed by, yet still seem to that listener to be the same
interval.
>
> Definition 2: _A listener's tolerance for error in tuning_ is the
> smallest value of that listener's tolerance for error in tuning an
> interval, taken over all intervals that listener can hear.
>
> Assumption: All intervals are played in a standard natural tuned
> acoustic timbre with which the listener is familiar.

Great, except for the assumption. Natural acoustic timbres would be
humans and birds and whales- guitar or synth, it's an artifact.

>
> Yep, I'm content that that ("character") describes something that
> some listeners hear.

Probably the most common way of hearing music, don't you think?

>
>
> > > Then, within the range of each of these intervals plus or
>>minus
> your tolerance, do you find simpler ratios that you might be
> assimilating them to?
> >
> > I think I assimilate to character families.
>
> I guess I should have written:
> "Within the range of each of these intervals plus or minus that
> listener's tolerance for error, are there objectively simpler
> ratios? If so, that listener might be assimilating them to those
> ratios."

I assume that must be true at a certain point, makes sense.
>
>
> > > Or in other words, does either interval lie in a basin of
> > > attraction of a simpler ratio?
> >
> > Thinking about it- I don't buy into "basin of attraction".
>
> Perhaps that means: It doesn't describe what you hear? Tell me,
>does
> the "fifthness" of intervals very near to 3/2 not tend to swamp
any
> other character they may express? For example, when you hear a
>73/49
> or a 74/49, does it sound like a fifth, or does it sound
>distinctly
> septimal?

Sometimes it does, like on one of the intervals Gene posted the
other day, a 722 cent fifth that sound very "thirdsy and minor" as I
responded- I had to stop and think to catch the fifth quality.

>You know, I think it might be a valuable experiment for
> you to generate a large random set of JI intervals in, say, the 99-
> odd-limit; then listen to them presented randomly (not more than
> about 20 at a sitting, to avoid fatigue effects) and assign each
to
> one of a predefined list of characters, eg,
> unison-like,
> octave-like,
> fifth-like,
> fourth-like,
> major-third-like,
> minor-sixth-like,
> minor-third-like,
> major-sixth-like,
> ...
> major-semitone-like,
> minor-semitone-like,
> ...
> Pyth.-comma-like,
> ...
> NONE.
>
> (Your list of characters might be quite different; this is just a
> suggested starting point.)

That's the same as my list, but I have the median intervals in their
as well. I haven't been methodical though.
>
> Then analyse the results to see whether the experiment supports
>your
> feeling that you are hearing these kinds of character; in
particular,
> which characters you strongly (95% of the time or better)
associate
> to ratios involving a particular prime in the numerator; and
> separately, which you strongly associate to ratios involving a
> particular prime in the denominator.

Well the theory came after the practice- I looked to see what might
be causing or contributing to the sensation and found the
relationship, which may be coincidence or simply another way of
expressing something else,of course.
>
> There'd be no point me doing such an experiment, I think, because
>I'm
> not at all sure I hear such character families.

You don't find 6ths more similar in some way to 3ds than to fifths?
Same experience, just not as loud. :-)

>But you, and others
> who do, would be suitable experimental subjects. And the effort
> might shed new light on what is going on in perception of musical
> intervals - that's one reason I stipulated the use of only natural
> tuned acoustic instruments, so the result might be relevant to
music.

I use synthesizers to make music. On the fretless guitar, I haven't
found anything that can't be explained by "pretty low ratios and 23
limit", but it's difficult to measure any finer than that, too.

> > > As a measure of ratio simplicity, log (n*d) probably works
>>>pretty
> > > well.
> > >
> > > For example, can we approximate 529/351 by, say 528/350 =
254/175?
> > > I haven't yet checked how many cents in either; but supposing
we
> > > could, does 254/175 lie within your tolerance of 529/351? If
so,
> > >you
> > > might mentally be assimilating the more complex ratio to the
> > >simpler
> > > one. Then 254/175 being a simpler ratio than 414/275 on the
log>
> > (n*d)
> > > measure, means you could just use n*d.
> >
> > That could very well be, it makes sense. In this case it doesn't
> > test my theory because the denominators octave-reduce only three
> > cents apart.
>
> What does that mean? I'd hazard a guess, but I doubt you could
> possibly mean what I think you mean here ... Please show me the
> numbers. (Or, the money would be fine. Really.)

Bet you've guessed, hahaha! Don't forget that I'm just trying to
explain why something I do works, not just cooking up a theory.
Finding the characters by ear, then trying to find the red threads.
No money, but you can judge for yourself if it works- the
song "Acacia" is the first one I've done entirely using a certain
approach, and "character families" are part of it.

http://www.zebox.com/bobro/music

> Not crap at all, Cameron; if anyone understood this stuff better
>than
> we yet do, you'd think they'd have explained it by now in a fairly
> publicly-accessible way. Personally, I think we've a very long
>way
> to go before we can effectively *describe* the varieties of human
> listening experience - much less *explain* them.

Well I agree, thanks for taking your time to answer so well!

-Cameron Bobro

> > > > This might go some way to answering the question you asked:
> > > > How big can the numbers be?
> > >
> > > The answer I'm proposing is "no bigger than your *personal*
> > > error tolerance in tuning allows".
> >
> > Harmonic entropy actually has a parameter similar to tolerance,
> > called "s". ...
>
> Yes, I noticed that, and you may recall, asked how to determine a
> partcular listener's value of s - without any really satisfactory
> answer that I remember.

I don't remember that, but basically s is the only free variable
in the model. It has to be experimentaly determined. Experiments
show it to be around 1%.

> > ... The smaller s is, the narrower all the bands are, but
> > they still aren't equal in width. ...
>
> Do they vary with interval in exactly the same way as the
> listener's tolerance?

That's the idea.

> Is there even any experimental evidence
> that tells us how they vary with interval?

Not that I'm aware of, but there might be.

> > ... S was found to vary between
> > listeners in psychoacoustics experiments.
>
> Where might I find reports of these experiments?

http://scitation.aip.org/getabs/servlet/GetabsServlet?
prog=normal&id=JASMAN000054000006001496000001

-Carl

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
> Hi Carl,

> > The other point I'd like to make is that a fixed tolerance might
> > not be the right thing. One key observation of Partch is that the
> > simpler ratios have a much stronger "pull" than the complex ones.
> > So the tolerance band that includes 3:2 should be larger than the
> > one that includes 254:175, and if they're in the same band the
band
> > should be called the 3:2 band.
>
> Yes, that makes sense. But could still be accommodated in a theory
> of the type I outlined. Do we have any experimental evidence of
how
> the tolerances vary with interval?

I think perhaps it does not need to be accomodated, because the wider
band is taken care of by the fact that the smaller intervals also
have a stronger "zone of repulsion" for small ratios. Hence if we
just, say, limit the numerator to be less than 20 or the product of
numberator and denominator to be less than 1000 or whatever along
those lines appeals to us, we get wider bands around the simpler
intervals for free. For a specific boundry line we might use the
mediant.

> > Where might I find reports of these experiments?
>
> http://scitation.aip.org/getabs/servlet/GetabsServlet?
> prog=normal&id=JASMAN000054000006001496000001

By the way, if anyone can access the full text of this,
I'd appreciate it being passed on to me. I only have a
hard copy.

-Carl

> http://scitation.aip.org/getabs/servlet/GetabsServlet?
> prog=normal&id=JASMAN000054000006001496000001

Or

http://tinyurl.com/yx6a9q

-Carl

Gene Ward Smith wrote:
>
> > > The other point I'd like to make is that a fixed tolerance
might not be the right thing. One key observation of Partch is that
the simpler ratios have a much stronger "pull" than the complex ones.
So the tolerance band that includes 3:2 should be larger than the one
that includes 254:175, and if they're in the same band the band
should be called the 3:2 band.

But not if the band is already called, say, The Rolling Stones ;-)

[Yahya]
> > Yes, that makes sense. But could still be accommodated in a
theory of the type I outlined. Do we have any experimental evidence
of how the tolerances vary with interval?

Please note I'm asking for real-world measurements of perceptual
phenomena ...

[Gene]
> I think perhaps it does not need to be accomodated, because the
wider band is taken care of by the fact that the smaller intervals
also have a stronger "zone of repulsion" for small ratios. ...

"Pull" & "repulsion" amount to a model of consonance that uses the
notion of force, some sort of dynamic model. Do you intend such
forces to be real and measurable, or only a convenient allegory?

> ... Hence if we just, say, limit the numerator to be less than 20
or the product of numberator and denominator to be less than 1000 or
whatever along those lines appeals to us, ...

That sounds pretty arbitrary to me.

> ... we get wider bands around the simpler intervals for free. For a
specific boundry line we might use the mediant.

Indeed, we might, but why should we? Will doing so bring us closer
to understanding what actually happens in human perception of
consonance? Or is this only a heuristic approach that paints us a
broad-brush picture, emphasising only that "smaller-numbered ratios
have more 'pull'"? As a starting point, I suppose it's OK, but it
doesn't really seem to advance our understanding greatly beyond
scholastic theories that classify all intervals simply into
consonances and dissonances.

Using the mediant (of two intervals) to define the boundary between
their respective ranges of "pull" is an interesting notion. How
might this work? For example:
between 4/3 and 3/2, the mediant is (4+3)/(3+2) = 7/5;
between 4/3 and 7/5, the mediant is (4+7)/(3+5) = 11/8;
between 4/3 and 11/8, the mediant is 15/11;
between 4/3 and 15/11, the mediant is 19/14;
between 11/8 and 15/11, the mediant is 26/19;
between 7/5 and 19/14, the mediant is 26/19;
between 11/8 and 19/14, the mediant is 15/11;
between 7/5 and 15/11, the mediant is 11/8;
and so on. We see that the mediant of any two of these terms is very
often (but not always) identical with another mediant we calculated
earlier; For example, the mediants of 11/8 with 9/7, 5/4 and 1/1 are
all equal to 4/3; however, the mediant of 11/8 with (the mediant of
5/4 and 1/1) is the mediant of 11/8 with 6/5, which is 17/13, not 4/3.

Inserting successive mediants between a note and its octave, we get:
a) 1/1---------------------------------------------------------2/1
b) 1/1-------------------------------------3/2-----------------2/1
c) 1/1-----------------4/3-----------------3/2-------5/3-------2/1
d) 1/1-------5/4-------4/3-------7/5-------3/2--8/5--5/3--7/4--2/1
e) 1/1--6/5--5/4--9/7--4/3-11/8--7/5-10/7--3/2- etc.

The maximum numerator N = max(n) and denominator D = max(d) in each
row both follow the Fibonnaci sequence: (1) 2 3 5 8 13 21 ..., as
does the maximum difference X = max(n-d).

To first approximation (case b), the intervals 1/1 and 2/1 have bands
that meet at 3/2; if correct, this means that any interval less than
a perfect fifth is "pulled" more strongly downwards - toward the
unison, rather than up toward the octave; whilst any interval greater
than a perfect fifth is "pulled" more strongly upwards - toward the
octave, rather than down toward the unison.

Similarly for each succeeding case. When we interpolate the fifth
3/2, it has a band which steals a portion from each of the two
previous bands - the lower part, from 4/3 to 3/2, from the unison
band; and the upper part, from 3/2 to 5/3, from the octave band. Any
interval that falls within the 3/2 band will be "pulled" more
strongly toward it than to either the unison or octave.

The insertion of the new interval, 3/2, into the previous set of
intervals {1/1, 2/1} *changes* their bands of attraction, reducing
them both to end at the new mediants 4/3 and 5/3. There is thus a
natural sequence of scales S1, S2, S3, ... Sn, ... formed by the
operation of taking mediants, each such scale being embedded in its
extension by its own mediants. The first four and a half such scales
are shown above at a) to e).

These considerations show us that the width of the band of attraction
of any interval is not fixed, but varies depending on what other
intervals we include in the analysis. Only in the context of a fixed
scale of just notes can we regard the attraction band's width as
effectively fixed.

(Also, a band of attraction is not necessarily a band of tuning error
tolerance, which is what I started talking about earlier in
the "exponent-limit and prime-affect" thread.)

For the sake of argument, let us say that a just note is _stronger
[weaker]_ than another if its interval ratio is, in some computable
sense, simpler [more complex] than that of the other. We might use
denominator, or n*d, or any of several other possible measures of
simplicity of ratios. (Aside: May I just note here that the n*d
measure handicaps higher intervals somewhat compared with lower
ones. Eg, 4/3 would be judged simpler than 5/3 on this measure, even
though both arrived in the same scale S3. Or is numerator more
important than denominator? If we feel that thirds and sixths are
about equally complex, is that a judgment that 5/4 and 5/3 are both
simpler than 7/5, and both less simple than 4/3?)

Also for the sake of argument, let us suppose that in every tempered
scale, each note is a tempered approximation to a just note, which is
a just interval above the unison; and just which just note that is,
is quite clear from the musical context. (The most telling clue
would be the density of notes in the octave, or equivalently, the
average step size.) On this basis - which probably holds true for a
lot of music, though not all - for each note n in the scale, we could
write just(n) for the corresponding just ratio.

We could then say, in any tempered scale, that one note n is
_stronger [weaker]_ than another note m if just(n) is _stronger
[weaker]_ than just(m).

Note that I wrote "if correct" above. Do the bands of attraction of
a pair of just intervals always meet in their mediant? On the face
of it, this does seem plausible. Surely we should be able to find
statistical evidence for or against this idea in a large corpus of
music, given some reasonable assumptions as to the nature of that
music? For example, the kind of thing played on your local "easy
listening" radio station would, we might guess, not fight too
greatly, on average, against the "pull" of the intervals of the scale
used.

Specifically, one would predict that:
(1) the number of weak notes that move _to_ a stronger note is
greater than the number of weak notes that move _to_ a weaker note
(other than itself (*));
(2) the number of weak notes that move _toward_ a stronger note is
greater than the number of weak notes that move _toward_ a weaker
note (other than itself (*));
(3) If a weak note n lies below [above] its corresponding just ratio;
that is, if n < just(n); and if just(n) is the mediant of the just
notes for two other stronger notes p and q in the scale, with p lower
than q; that is, if just(n) = mediant(just(p),just(q)) where p and q
are scale notes stronger than n, with p < q; then the number of
occurrences of n that move toward p will be greater [less] than the
number that move toward q.

In terms of the "dynamical model" approach, of interpreting
consonance as a sort of gravitational attraction, or "pull", and even
possibly of "repulsion", I must say that the predictions of this
sketch do not include any of the effects of _momentum_. Thus it is a
static, rather than a dynamic model. We observe all kinds of
analogues to momentum in music, from the simple run, ascending or
descending; to motivic repetition and sequence; to cadential motion
resolving discords into concords; to completely non-tonal phenomena
such as metre and rhythm.

(*) There may also be a "local tonic" or "tonic shift" effect, where
even a weak note may simply be repeated more often than it moves to a
stronger note. This might also be treated as a "momentum" effect.

However, with the above cautions, we should still find something like
the tendencies noted in points (1) to (3) above - at least in
the "easy listening", "light classic" and "elevator music" categories.

If there really is a "mediant effect" in the perception of consonance
or at work in music, it would be interesting to speculate why this
might be so. Is there perhaps something in the mammalian, or more
specifically, human perceptual system that generates mediants as
artefacts of perception?

Still, until we can actually demonstrate a "mediant effect", it would
be premature to try to explain it ...

Regards,
Yahya

Carl Lumma wrote:
>
> > > > > This might go some way to answering the question you asked:
> > > > > How big can the numbers be?
> > > >
> > > > The answer I'm proposing is "no bigger than your *personal*
> > > > error tolerance in tuning allows".
> > >
> > > Harmonic entropy actually has a parameter similar to tolerance,
> > > called "s". ...
> >
> > Yes, I noticed that, and you may recall, asked how to determine a
> > partcular listener's value of s - without any really satisfactory
> > answer that I remember.
>
> I don't remember that, but basically s is the only free variable
> in the model. It has to be experimentaly determined. Experiments
> show it to be around 1%.
>
> > > ... The smaller s is, the narrower all the bands are, but
> > > they still aren't equal in width. ...
> >
> > Do they vary with interval in exactly the same way as the
> > listener's tolerance?
>
> That's the idea.
>
> > Is there even any experimental evidence
> > that tells us how they vary with interval?
>
> Not that I'm aware of, but there might be.
>
> > > ... S was found to vary between
> > > listeners in psychoacoustics experiments.
> >
> > Where might I find reports of these experiments?
>
> http://scitation.aip.org/getabs/servlet/GetabsServlet?
> prog=normal&id=JASMAN000054000006001496000001

Hi Carl,

That's just an article abstract, which reports a model of pitch
determination of noisy complex tones which has one free parameter -
the variance. Without access to the article, I'm unable to see the
relevance to the current questions, or even an attribution of the
variance as between-listeners. From the abstract, I rather got the
impression the variance was betweem-stimuli.

Regards,
Yahya

> > > Where might I find reports of these experiments?
> >
> > http://scitation.aip.org/getabs/servlet/GetabsServlet?
> > prog=normal&id=JASMAN000054000006001496000001
>
> Hi Carl,
>
> That's just an article abstract, which reports a model of pitch
> determination of noisy complex tones which has one free parameter -
> the variance. Without access to the article, I'm unable to see the
> relevance to the current questions, or even an attribution of the
> variance as between-listeners. From the abstract, I rather got the
> impression the variance was betweem-stimuli.
>
> Regards,
> Yahya

Hrm, that may be the wrong cite. Sorry. I'll have to dig
through some papers...

-Carl

I was trying to write a reply, and it automatically posted itself.

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> > I think perhaps it does not need to be accomodated, because the
> wider band is taken care of by the fact that the smaller intervals
> also have a stronger "zone of repulsion" for small ratios. ...
>
> "Pull" & "repulsion" amount to a model of consonance that uses the
> notion of force, some sort of dynamic model. Do you intend such
> forces to be real and measurable, or only a convenient allegory?

It's merely figurative language for a fact of number theory.

> > ... Hence if we just, say, limit the numerator to be less than 20
> or the product of numberator and denominator to be less than 1000
or
> whatever along those lines appeals to us, ...
>
> That sounds pretty arbitrary to me.

What would not sound arbitrary to you?

> We see that the mediant of any two of these terms is very
> often (but not always) identical with another mediant we calculated
> earlier; For example, the mediants of 11/8 with 9/7, 5/4 and 1/1
are
> all equal to 4/3;

You need to have the two intervals be contiguous on the Farey
sequence, so that a/b and c/d are such that |ad-bc| = 1. But
|11*4-5*8| = 4, |11*5-6*8| = 7, |11*1-8*1| = 3, so none of these are
contiguous.
> These considerations show us that the width of the band of
attraction
> of any interval is not fixed, but varies depending on what other
> intervals we include in the analysis.

This is why I introduced the constraints you called arbitary; you
seem to be doing much the same thing yourself now.

> On this basis - which probably holds true for
a
> lot of music, though not all - for each note n in the scale, we
could
> write just(n) for the corresponding just ratio.

OK, but it is easier mathematically to do things the other way
around: for a rational number q in some prime limit, tempered(q) is
an approximation. Do this regularly and you have a tuning map.

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

>
> I've got another theory. First, what is your *personal* tolerance
> for error in tuning: 2 cents? half a cent? one-third of a cent?
>

I've kept your question in mind since you asked and it seems clear
to me that measuring in quantity is a dubious proposition, as my
tolerance ranges from less than a cent to over 20 cents. I don't
listen for size, but character. In other words, listening to
different intervals sequentially, not as diads but with occaisional
reference (as much as needed) to a ground tone, two almost
identical, in sheer size, intervals may sound jarringly different,
and two quite far from each other "the same".

> Then, within the range of each of these intervals plus or minus
>your
> tolerance, do you find simpler ratios that you might be
>assimilating
> them to?

Now I can say, definitely not, at least in tuning with musical goals
in mind, which is the only kind of tuning I have time to bother with.
Assimilating to simpler intervals simply doesn't explain why I
sometimes choose a very complex ratio over a far simpler ratio two
cents away.

-Cameron Bobro

-Cameron Bobro

Hi Mr Bobro,

Cameron Bobro wrote:
>
> --- In tuning@yahoogroups.com, "yahya_melb" <yahya@> wrote:
>
> >
> > I've got another theory. First, what is your *personal*
tolerance for error in tuning: 2 cents? half a cent? one-third of a
cent?
> >
>
> I've kept your question in mind since you asked and it seems clear
> to me that measuring in quantity is a dubious proposition, as my
> tolerance ranges from less than a cent to over 20 cents. I don't
> listen for size, but character. In other words, listening to
> different intervals sequentially, not as diads but with occaisional
> reference (as much as needed) to a ground tone, two almost
> identical, in sheer size, intervals may sound jarringly different,
> and two quite far from each other "the same".

I take your point. I reckon much the same happens for me, if not as
widely (probably from half a cent to around 6 cents); but I'm at a
loss to explain that "character", except as something timbral.

> > Then, within the range of each of these intervals plus or minus
your tolerance, do you find simpler ratios that you might be
assimilating them to?
>
> Now I can say, definitely not, at least in tuning with musical
goals in mind, which is the only kind of tuning I have time to bother
with.

OK.

> Assimilating to simpler intervals simply doesn't explain why I
sometimes choose a very complex ratio over a far simpler ratio two
cents away.

Do you know what *does* explain it?

Regards,
Yahya

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> I take your point. I reckon much the same happens for me, if not
>as
> widely (probably from half a cent to around 6 cents); but I'm at a
> loss to explain that "character", except as something timbral.

I forgot to add that I just took a couple of "scientific" tests,
with sine generators, headphones, and oscilliscopes, at the Phaeno
center, where I was doing some shows. I could say, dishonestly, that
my tolerance is zero, and "prove" that with the tone matching test
(different tones in both ears, then you turn a big knob to make them
match), but that would be misleading (although I bet people use such
results to sport their prowess). That is a highly artificial test- I
highly doubt I can do this with a stringed instrument, even if I
wanted to. The slow "creamy" sound of almost-exact sounds better
when matching pitches, to me; I had to go beyond what sounded best
to make the "disappear" effect of an exact match).

> Do you know what *does* explain it?

Now that I've been digging around a bit trying to find an
explanation, it might be something like Partch's "numerary nexus".
Something along those lines- I'm calling it character families
because that's what it sounds like to me, and I want to stay away
from magic numbers.

I had overlooked Gene's post to the effect that we must try with
irrational numbers and of course he's right about that. However, as
I've said before, there must be a "don't be silly" proximity- how do
I know that the irrational interval doesn't lie within .2 cents of a
distinct rational interval of the right character family? This is
one reason I'm keen on the idea of character families, because they
would reduce into what seems to me reasonable regions, not magic
numbers.

Perhaps it is timbre related, as you say. The answer may lie within
say the first dozen partials or so, and once the fundamentals are no
longer at a simple ratio we may seek simple ratios for, say, the
third or seventh partials? That seems to fit in with one of my
first posts here, where I noted that sometimes I have to open the
filter on a synth sound to make it sound more consonant.

If the answer lies above the fundamental, it would be VERY timbre
dependent, of course. So I must now test with different timbres- a
square and a saw are the obvious starting places because if the
effect simply disappears, or is obviously stronger, with the square,
it's all clearly related to partials above the fundamental.

To find the time...

-Cameron Bobro