Fried Alaska

If we define the octave-fifth beat ratio as

(2f - 3)/(o - 2)

where f is the fifth and o is the octave, and if we use a repeating
alaska pattern of scale steps with eight steps of size "a", satisfying

16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0

and four steps of size "b" satisfying

256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8-
18432*b^10+4096*b^12 = 0

then we get an alaska-style scale with eight octave-fifth brats equal
to 2 and four equal to 1; Fried Alaska:

! alafried.scl
Fried alaska, with octave-fifth brats of 1 and 2
12
!
98.867788
197.735576
299.074366
397.942153
496.809941
598.148731
697.016520
795.884307
897.223099
996.090887
1094.958674
1196.297466

>If we define the octave-fifth beat ratio as
>
>(2f - 3)/(o - 2)
>
>where f is the fifth and o is the octave,

f and o are in cents? Could you give the derrivation
of this formula?

>if we use a repeating alaska pattern of scale steps with eight steps
>of size "a", satisfying
>
> 16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0
>
>and four steps of size "b" satisfying
>
>256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8-
>18432*b^10+4096*b^12 = 0

Honestly, I don't know how you do it, Gene.

>then we get an alaska-style scale with eight octave-fifth brats equal
>to 2 and four equal to 1; Fried Alaska:
>
>! alafried.scl
>Fried alaska, with octave-fifth brats of 1 and 2
>12
>!
>98.867788
>197.735576
>299.074366
>397.942153
>496.809941
>598.148731
>697.016520
>795.884307
>897.223099
>996.090887
>1094.958674
>1196.297466

This is fantastic. It's very close to Alaska VI, but with better
bad thirds, better fifths, and even slightly better octaves. The
good thirds suffer, but as 10ths they'll still be doing quite
well. Fantastic!

Now, to test to see if the brats make a difference. I'll report
back...

-Carl

>Now, to test to see if the brats make a difference. I'll report
>back...

Well, it's hard to test just the brats, since this is slightly
different than any non-synched alaska tuning. But after a quick
test with a Reed organ patch, I don't prefer this to Alaska VI.

I can definitely hear a difference between vallotti and synced
vallotti, which are quite close tuning-wise. But I think the
best thing would be to render a midi file in Audio Compositor
that compares synched and non-synched triads in various
inversions. I may do this at some point.

-Carl

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

> >If we define the octave-fifth beat ratio as
> >
> >(2f - 3)/(o - 2)
> >
> >where f is the fifth and o is the octave,
>
> f and o are in cents? Could you give the derrivation
> of this formula?

They are frequency ratios; if we have a "trine" of 1, f, o, then
the beats between f and 1 are 2f-3 and between o and 1 o-2, the above
is therefore a beat ratio. There are five others, since we also have
beats from f to o; if r is the above ratio the others are given by
r, 2r - 3, 3/r - 2 and their reciprocals.

> >if we use a repeating alaska pattern of scale steps with eight steps
> >of size "a", satisfying
> >
> > 16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0
> >
> >and four steps of size "b" satisfying
> >
> >256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8-
> >18432*b^10+4096*b^12 = 0
>
> Honestly, I don't know how you do it, Gene.

Since you have Maple, you could do it too, using the resultant function.

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

> I can definitely hear a difference between vallotti and synced
> vallotti, which are quite close tuning-wise.

If you can tell the difference between werckmeister and
sync-werckmeister, which are nearly identical, it would definately be
a plus for the theory that syncing the beats matters.

>> >If we define the octave-fifth beat ratio as
>> >
>> >(2f - 3)/(o - 2)
>> >
>> >where f is the fifth and o is the octave,
>>
>> f and o are in cents? Could you give the derrivation
>> of this formula?
>
>They are frequency ratios; if we have a "trine" of 1, f, o, then
>the beats between f and 1 are 2f-3 and between o and 1 o-2, the above
>is therefore a beat ratio.

Ah! That's sooo simple. Thanks!

>> >if we use a repeating alaska pattern of scale steps with eight steps
>> >of size "a", satisfying
>> >
>> > 16*a^12+72*a^10+81*a^8-64*a^7-120*a^5-32 = 0
>> >
>> >and four steps of size "b" satisfying
>> >
>> >256+864*b^3+6561*b^4-384*b^5-23328*b^6-512*b^7+31104*b^8-
>> >18432*b^10+4096*b^12 = 0
>>
>> Honestly, I don't know how you do it, Gene.
>
>Since you have Maple, you could do it too, using the resultant function.

Yeah, but how'd you get those polynomials?

-Carl

>> I can definitely hear a difference between vallotti and synced
>> vallotti, which are quite close tuning-wise.
>
>If you can tell the difference between werckmeister and
>sync-werckmeister, which are nearly identical, it would definately be
>a plus for the theory that syncing the beats matters.

These temperaments are besides the point. We need a series of
triads tuned as near as they can be, with a selection of brats.
If you provide them in cents in root position, I'll make an
audio sequence and post it.

-Carl

>These temperaments are besides the point. We need a series of
>triads tuned as near as they can be, with a selection of brats.
>If you provide them in cents in root position, I'll make an
>audio sequence and post it.

Oh, and don't publish the brats. Just number the triads.

-Carl

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

> These temperaments are besides the point. We need a series of
> triads tuned as near as they can be, with a selection of brats.
> If you provide them in cents in root position, I'll make an
> audio sequence and post it.

I['m not sure what you mean--different tunings can have the same
brat, and for any given brat, you can get arbitrarily close to JI.

>> These temperaments are besides the point. We need a series of
>> triads tuned as near as they can be, with a selection of brats.
>> If you provide them in cents in root position, I'll make an
>> audio sequence and post it.
>
>I'm not sure what you mean--different tunings can have the same
>brat,

Of course. And they should be included in the mix.

>and for any given brat, you can get arbitrarily close to JI.

Oh yeah? Then brats are nonsense, I say.

I want two chords, with brats of 2 and 13/8, each 1 cent RMS
from 3:4:5 to the nearest .1 cent RMS.

-Carl

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

> >and for any given brat, you can get arbitrarily close to JI.
>
> Oh yeah? Then brats are nonsense, I say.

wha???

>> >and for any given brat, you can get arbitrarily close to JI.
>>
>> Oh yeah? Then brats are nonsense, I say.
>
>wha???

At the least it means we'd need to consider the error as well
as the brat -- brats alone would not be reliable.

-Carl

>>>>and for any given brat, you can get arbitrarily close to JI.
>>>
>>>Oh yeah? Then brats are nonsense, I say.
>>
>>wha???
>
>At the least it means we'd need to consider the error as well
>as the brat -- brats alone would not be reliable.

Ironically, you use this sort of reasoning against prime limit
all the time.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>>>and for any given brat, you can get arbitrarily close to JI.
> >>>
> >>>Oh yeah? Then brats are nonsense, I say.
> >>
> >>wha???
> >
> >At the least it means we'd need to consider the error as well
> >as the brat -- brats alone would not be reliable.
>
> Ironically, you use this sort of reasoning against prime limit
> all the time.

you're replying to yourself here.

what sort of reasoning? let's see the analogy fleshed out.

>> >At the least it means we'd need to consider the error as well
>> >as the brat -- brats alone would not be reliable.
>>
>> Ironically, you use this sort of reasoning against prime limit
>> all the time.
>
>you're replying to yourself here.

Yeah...

>what sort of reasoning? let's see the analogy fleshed out.

There are ratios of low prime limit that are quite dissonant.
There are chords indistinguishable from just with very high brats.

I dunno, the sort of reasoning that looks at the function over
the pitch continuum and sees if there are anomalies.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >At the least it means we'd need to consider the error as well
> >> >as the brat -- brats alone would not be reliable.
> >>
> >> Ironically, you use this sort of reasoning against prime limit
> >> all the time.
> >
> >you're replying to yourself here.
>
> Yeah...
>
> >what sort of reasoning? let's see the analogy fleshed out.
>
> There are ratios of low prime limit that are quite dissonant.
> There are chords indistinguishable from just with very high brats.

i thought were were talking about well-temperaments with low, but
rational, brats? you obviously don't care what the brat is for any
chord close enough to just -- it's the significantly tempered ones
where brats may or may not be meaningful. also, it's not the lowness
of the brat, it's the simplicity of the ratio.

>i thought were were talking about well-temperaments with low,
>but rational, brats?

We were, but I suggested we compare bare chords instead of
temperaments, because that's what the assumption about the larger
tunings is based on.

>you obviously don't care what the brat is for any chord close enough
>to just -- it's the significantly tempered ones where brats may or
>may not be meaningful.

Apparently so.

>also, it's not the lowness of the brat, it's the simplicity of t

Right; I've just been saying lowness.

By simplicity, don't we mean something like the Van Eck widths?

-Carl

>>you obviously don't care what the brat is for any chord close enough
>>to just -- it's the significantly tempered ones where brats may or
>>may not be meaningful.
>
>Apparently so.

Also, if we replace "just" with "interval x" in Gene's statement,
brats are completely useless without some sort of tolerance adjustment,
like we use for odd limit.

-Carl

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> i thought were were talking about well-temperaments with low, but
> rational, brats? you obviously don't care what the brat is for any
> chord close enough to just -- it's the significantly tempered ones
> where brats may or may not be meaningful. also, it's not the lowness
> of the brat, it's the simplicity of the ratio.

A brat of, say, -1 close to just will simply beat very slowly in sync.
As for your second point, infinity happens to be a very nice brat.

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

> By simplicity, don't we mean something like the Van Eck widths?

That's tough to answer without knowing what a Van Eck width is, but
the rules of simplicity are these:

(1) q and -q, 1/q, and -1/q are equally simple. In particular, 0 and
infinity are equally simple.

(2) To judge how simple q is, you need to look at how simple 5/(3-2q)
and (3-2q)/5q are as well.

(3) Low numerators and denominators are better than high ones; bearing
in mind that infinity = 1/0 counts as low.

>> By simplicity, don't we mean something like the Van Eck widths?
>
>That's tough to answer without knowing what a Van Eck width is,

I found gcdb by the way, but have no idea how it works.

The Van Eck width of ratio Ri is log(R(i-1))-log(R(i+1)), where
Ri is, say, the ith ratio in a Farey series of order n.

Unfortunately, I guess these widths just shrink to nothing as
n goes to infinity. The harmonic entropy based on them, however,
converges to a finite value.

>but the rules of simplicity are these:
>
>(1) q and -q, 1/q, and -1/q are equally simple. In particular, 0 and
>infinity are equally simple.
>
>(2) To judge how simple q is, you need to look at how simple 5/(3-2q)
>and (3-2q)/5q are as well.
>
>(3) Low numerators and denominators are better than high ones;
>bearing in mind that infinity = 1/0 counts as low.

Since we have no idea what's supposed to make one brat better than
another, I suppose this is fine.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> I found gcdb by the way, but have no idea how it works.

One attempt to automate the judgment of goodness is all it is.

> The Van Eck width of ratio Ri is log(R(i-1))-log(R(i+1)), where
> Ri is, say, the ith ratio in a Farey series of order n.

Great! Now someone should be able to write down a formula for harmonic
entropy in terms of the function VE(r).

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> By simplicity, don't we mean something like the Van Eck widths?
> >
> >That's tough to answer without knowing what a Van Eck width is,

it's actually quite similar to tenney height, as i showed on the
harmonic entropy list.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > I found gcdb by the way, but have no idea how it works.
>
> One attempt to automate the judgment of goodness is all it is.
>
> > The Van Eck width of ratio Ri is log(R(i-1))-log(R(i+1)), where
> > Ri is, say, the ith ratio in a Farey series of order n.

or a "tenney series", or whatever.

> Great! Now someone should be able to write down a formula for
harmonic
> entropy in terms of the function VE(r).

for a logarithmic interval q, the unnormalized probability of hearing
it as ratio r, P(q,r), is

UP(q,r)=e^-((log(r)-q)^2/2s) [for bell-curve entropy]

or

UP(q,r)=e^-(|(log(r)-q)|/s) [for Vos-curve entropy]

where s parameterizes one's hearing resolution.

the probability is simply the normalized probability divided by the
sum of the unnormalized probabilities:

P(q,r)=UP(q,r)/SUM(UP(q,r))

where the sum takes r over all possible values within the farey,
tenney, whatever series in question.

then the harmonic entropy of interval q, HE(q), is

SUM[P(q,r)*log(P(q,r))]
r

ok?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> ok?

No; how do you sum over row infinity of the Farey sequence?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > ok?
>
> No; how do you sum over row infinity of the Farey sequence?

first of all, carl and i both had errors. carl's definition was wrong
because the width goes from mediant to mediant. my probability
definition was wrong because i only included the height term but
forgot to multiply by the aforementioned width to get the area under
the curve!

now, row infinity? it's the farey, or mann, or tenney series of order
n. n and s are the two parameters of the harmonic entropy function,
plus you get to choose farey/mann/tenney/etc., and you get to choose
bell/Vos.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> now, row infinity? it's the farey, or mann, or tenney series of
order
> n. n and s are the two parameters of the harmonic entropy function,
> plus you get to choose farey/mann/tenney/etc., and you get to
choose
> bell/Vos.

Does harmonic entropy seem to converge to a continuous function as n
goes to infinity?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > now, row infinity? it's the farey, or mann, or tenney series of
> order
> > n. n and s are the two parameters of the harmonic entropy
function,
> > plus you get to choose farey/mann/tenney/etc., and you get to
> choose
> > bell/Vos.
>
> Does harmonic entropy seem to converge to a continuous function as
n
> goes to infinity?

it's continuous for any n.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > now, row infinity? it's the farey, or mann, or tenney series of
> order
> > n. n and s are the two parameters of the harmonic entropy
function,
> > plus you get to choose farey/mann/tenney/etc., and you get to
> choose
> > bell/Vos.
>
> Does harmonic entropy seem to converge to a continuous function as
n
> goes to infinity?

as n goes to infinity, the "shape" seems to converge, but it gets
taller and flatter. if we had some suitable way to correct for the
tallness and flatness as a function of n, we might be able to define
a function which indeed converges to a limit as n goes to infinity.
this is my hope, as i communicated to you some time ago.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > >
> > > now, row infinity? it's the farey, or mann, or tenney series of
> > order
> > > n. n and s are the two parameters of the harmonic entropy
> function,
> > > plus you get to choose farey/mann/tenney/etc., and you get to
> > choose
> > > bell/Vos.
> >
> > Does harmonic entropy seem to converge to a continuous function
as
> n
> > goes to infinity?
>
> as n goes to infinity, the "shape" seems to converge, but it gets
> taller and flatter. if we had some suitable way to correct for the
> tallness and flatness as a function of n, we might be able to
define
> a function which indeed converges to a limit as n goes to infinity.
> this is my hope, as i communicated to you some time ago.

i actually made some inroads into acheiving this, but without any
sort of mathematical insight or proof. i recommend you spend some
time looking over the harmonic entropy archives, they are quite short
compared to those of other lists. and let's continue the discussion
there, shall we?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> > Does harmonic entropy seem to converge to a continuous function
as
> n
> > goes to infinity?
>
> it's continuous for any n.

Yes, but does it converge uniformly for increasing n?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> as n goes to infinity, the "shape" seems to converge, but it gets
> taller and flatter. if we had some suitable way to correct for the
> tallness and flatness as a function of n, we might be able to
define
> a function which indeed converges to a limit as n goes to infinity.
> this is my hope, as i communicated to you some time ago.

Ah. Somehow I had the idea that you were claiming it did, and I
didn't see it.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> i actually made some inroads into acheiving this, but without any
> sort of mathematical insight or proof. i recommend you spend some
> time looking over the harmonic entropy archives, they are quite
short
> compared to those of other lists. and let's continue the discussion
> there, shall we?

Why is there such a list?

>Why is there such a list?

Rather than ask a question like that, why not accept it with
the rest of these miserable lists, and go there, where I've
already forwarded this thread, so as to commingle it with the
extremely valuable archives there?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > i actually made some inroads into acheiving this, but without any
> > sort of mathematical insight or proof. i recommend you spend some
> > time looking over the harmonic entropy archives, they are quite
> short
> > compared to those of other lists. and let's continue the
discussion
> > there, shall we?
>
> Why is there such a list?

it began quite some time before this list began. there were quite a
few harmonic entropy postings on the tuning list, which made certain
people very upset. so it became its own list. much later, the "big
split" occurred. it was suggested that harmonic_entropy be combined
with this list, but i saw no way of doing that.