Yahoo Tuning Groups Ultimate Backup tuning-math Farey height and Farey norms, intro

Sometimes, Tenney height is used to bound intervals by complexity, and
other times, odd-limit is used. There are cases when the former is
more appropriate, and cases when the latter is more appropriate. Farey
height is an alternative to Tenney height which represents the latter
way of thinking, and which will enable us to define:

1) a "Farey norm" on interval space representing the Farey height of
any interval and a corresponding dual norm on vals and tuning maps,
2) a class of "Farey optimal", or FOP tunings which minimize
Farey-weighted error over all intervals,
3) an easily computable L2-based approximation to the Farey norm
called the "Farey-Euclidean" or "FE" norm, and a corresponding FE
tuning,
4) a completely generalized class of Fp norms paralleling the Tp norms
we've talking about for the past week,
5) quotient norms corresponding to the usual odd-limit,
3/1-equivalent-limit, etc as special cases.

To accomplish all this, we drop the initial assumption of octave
equivalence and define the Farey height of any interval as max(n,d)
(or log(max(n,d)); this is basically the "integer-limit" of any
interval. To do this treats 2/1 as any other vector and makes things
much more straightforward. Then, when we want to impose
octave-equivalence and look at odd-limit, we can easily define Kees
expressibility as a quotient norm by modding by the 2/1 vector (or, if
you like, to define a seminorm on interval space). The same applies
for settings where you want 3/1 to be equivalent instead of 2/1.

Aside from using this as a launch pad for getting different seminorms,
there may be some situations where integer-limit is itself useful. For
instance, odd-limit is a useful way of bounding intervals in
situations where you want 5/3 before 15/1, and as such tends to give
you intervals which combine well to form simple chords. While
octave-equivalence is probably a decent rule of thumb for chord
intonation for the 5-limit, for higher-limit chords there are
certainly cases where it starts to get trickier: for instance, one
might compare 4:7:9:11 to 8:9:11:14, or 4:7:11:13 to 8:11:13:14, or
2:3:5:7:11:17 to 16:17:20:22:24:28 to perhaps find cases where octave
transposition makes a difference in the ways that chords are intoned.
So you can think of integer-limit as being like odd-limit in that it
still tends to give you intervals that combine well to make simple
chords, but which respects these differences in octave transposition.

Another interesting property of Farey height is that the following
identity holds:

max(n,d) = sqrt(n*d * max(n/d, d/n))

So if n*d is the complexity of an interval, and max(n/d, d/n) is its
span (both in linear-frequency) units, then the Farey height of any
interval is sqrt(complexity*span), e.g. the geometric mean of
complexity and span. Even more interesting is how this works out
logarithmically, as Keenan pointed out:

log(max(n,d)) = (log(n*d) + |log(n/d)|)/2

So identifying log(n*d) as complexity and log(n/d) as span in
log-frequency units, then the (log) Farey height of any interval is
(complexity+span)/2, e.g. the arithmetic mean of complexity and span.
So if you care about the span of an interval and feel that extremely
wide intervals should be penalized, this may be musically relevant for
you. If you don't care about span, then for you, this equates to a
simple note that many of the expressions that can be derived from this
Farey norm will automagically tend to be functions of the complexity
and span of an interval, but for you people this will be just a
non-musical coincidence. (We'll use this later to define a class of
highly generalized norms for which Tenney, Farey, TE, Kees, min(n,d),
etc are all special cases.)

This is a good chunk of information for now. I'll be back with part 2
shortly to explain how to define this in terms of monzo coordinates.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > 1) a "Farey norm" on interval space representing the Farey height of
> > any interval and a corresponding dual norm on vals and tuning maps
>
> Since Farey had nothing whatever to do with this height function, I really don't see that it's a good idea to hang his name on it. If you apply Weil height to positive rationals, this is what you get, so one solution would be to call it Weil height.

I just googled this idea, and these authors do exactly that:

http://www.math.rochester.edu/people/faculty/ttucker/Papers.dir/Equi.pdf

I say let's call it Weil height.

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Aug 16, 2012 at 1:50 AM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> I just googled this idea, and these authors do exactly that:
>
> http://www.math.rochester.edu/people/faculty/ttucker/Papers.dir/Equi.pdf
>
> I say let's call it Weil height.

OK, before we switch names, can we please confirm that Weil height
displays the correct behavior? For instance, the whole point of the
function I'm describing is that it also defines a norm, meaning that
|| |0.5 0 0> || = 0.5 * || |1 0 0> ||, and this behavior is essential
for defining things like temperamental complexity and using
Hahn-Banach and such. So if the absolute logarithmic Weil height
function is W(n/d) = log(max(|n|,|d|)), then does the generalized
version for algebraic number fields have it so that W(sqrt(2)) = 1/2 *
W(2)?

The whole point of Weil height is apparently to use it for algebraic
number fields, so it should be possible to answer this question, at
least for the above example by using the field Q(sqrt(2)). I've been
trying to figure out what the answer is, but I don't know enough about
p-adic numbers and what a "place" is to unpack the definition.

If it does match, then I'm up for changing it. If it doesn't, though,
then we shouldn't call this Weil height or it's going to confuse the
living hell out of everyone.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> OK, before we switch names, can we please confirm that Weil height
> displays the correct behavior? For instance, the whole point of the
> function I'm describing is that it also defines a norm, meaning that
> || |0.5 0 0> || = 0.5 * || |1 0 0> ||, and this behavior is essential
> for defining things like temperamental complexity and using
> Hahn-Banach and such. So if the absolute logarithmic Weil height
> function is W(n/d) = log(max(|n|,|d|)), then does the generalized
> version for algebraic number fields have it so that W(sqrt(2)) = 1/2 *
> W(2)?

Yes it does; equivalently, for arithmetic Weil height, H(sqrt(2)) = sqrt(2).

> The whole point of Weil height is apparently to use it for algebraic
> number fields, so it should be possible to answer this question, at
> least for the above example by using the field Q(sqrt(2)). I've been
> trying to figure out what the answer is, but I don't know enough about
> p-adic numbers and what a "place" is to unpack the definition.

I could unpack it for you, but rather than starting a tutorial on algebraic number theory, I suggest using an alternative definition of Weil height. Suppose x is an algebraic number, with minimal polynomial a0x^n + ... + an. Suppose the product of those roots with absolute value greater than 1 is P. Then H(x) = |a0P|^(1/n). In your example, a0=1, P=-2, and n=2, so H(sqrt(2)) = |-2|^(1/2) = sqrt(2).

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Aug 16, 2012 at 1:46 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> I could unpack it for you, but rather than starting a tutorial on
> algebraic number theory, I suggest using an alternative definition of Weil
> height. Suppose x is an algebraic number, with minimal polynomial a0x^n +
> ... + an. Suppose the product of those roots with absolute value greater
> than 1 is P. Then H(x) = |a0P|^(1/n). In your example, a0=1, P=-2, and n=2,
> so H(sqrt(2)) = |-2|^(1/2) = sqrt(2).

It works but seems like random magic to me.

The given definition for H(x) is that I'm supposed to sum log
max(|x|_v, 1) over all the absolute values v, right? And the different
"absolute values" just refers to the different p-adic norms, and then
the one ordinary absolute value, yes? So if | · |_r is the ordinary
absolute value, then for 5/3, I get

log(max(|5/3|_3,1)) + log(max(|5/3|_5,1)) + log(max(|5/3|_r,1))

and |5/3|_3 = 3, |5/3|_5 = 1/5, and |5/3|_r = 5/3, so I get

log(max(3,1)) + log(max(1/5,1)) + log(max(5/3,1)) = log(3) + log(1) +
log(5/3) = log(3) + log(5/3) = log(5)

Right? I still don't know what a "place" is, but this seems to work out.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Aug 16, 2012 at 3:49 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> log(max(3,1)) + log(max(1/5,1)) + log(max(5/3,1)) = log(3) + log(1) +
> log(5/3) = log(3) + log(5/3) = log(5)
>
> Right? I still don't know what a "place" is, but this seems to work out.

OK, it seems obvious now. So in music-theoretic terms, for an
unweighted monzo |a b c ...>, the formula is

1) Set all positive coordinates in the monzo to 0.
2) Get the T1 norm of the resulting modified monzo.
3) Get the span of the original unmodified monzo in units of "octaves".
4) Add them up.

So for 5/3, which is |0 -1 1>, this means you get
1) |0 -1 1> becomes |0 -1 0>
2) || |0 0 1> ||_T1 = log2(3)
3) span(|0 -1 1>) = log2(5/3)
4) log2(3) + log2(5/3) = log2(5)

So then the algebro-number-theoretic version of the thing we were
talking about on IRC, the identity that Keenan "won the race" to
figure out, is

H(x) = sum_v (|log(|x|_v)|_r)/2, where | · |_r is the Archimedean
absolute value. So for any rational p/q, this gives you H(p/q) =
(log(p*q) + |log(p/q)|)/2.

The definition above is notable because if you take it over all
absolute values OTHER than the Archimedean one, you get Tenney height,
and if you include the Archimedean one, you get Weil height. Or well,
more specifically, you get Tenney height/2.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Aug 16, 2012 at 6:20 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > Right? I still don't know what a "place" is, but this seems to work out.
>
> http://en.wikipedia.org/wiki/Absolute_value_(algebra)
> http://en.wikipedia.org/wiki/Discrete_valuation
>
> If you look at the concept for algebraic function fields, you can see a
> "place" actually is a place. Unfortunately, Wikipedia's article on algebraic
> function fields is crap.

OK, I see now. So given a place, which is an equivalence class of
absolute values, how do I actually evaluate a rational at that place?
Which absolute value in the equivalence class do I use?

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > OK, I see now. So given a place, which is an equivalence class of
> > absolute values, how do I actually evaluate a rational at that place?
> > Which absolute value in the equivalence class do I use?
>
> For any algebraic number field, or for Q? For extension fields, there are two ways people sometimes do things.
>

I can't seem to find a good explanation of this stuff, but I think the better method is the one which woks well with Haar measure and which works with extensions of Qp instead of treating the primes of a number field just like those of Q. So, x^2-2 is irreducible over Q2, and so you take |sqrt(2)|_2 = 1/sqrt(2), not 1/2. You take the 1/nth power of the valuation of the norm term, which is 1/2, so yoi get sqrt(1/2).

🔗Mike Battaglia <battaglia01@gmail.com>

On Thu, Aug 16, 2012 at 8:30 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> I can't seem to find a good explanation of this stuff, but I think the
> better method is the one which woks well with Haar measure and which works
> with extensions of Qp instead of treating the primes of a number field just
> like those of Q. So, x^2-2 is irreducible over Q2, and so you take
> |sqrt(2)|_2 = 1/sqrt(2), not 1/2. You take the 1/nth power of the valuation
> of the norm term, which is 1/2, so yoi get sqrt(1/2).

This is the alternative definition you gave, right?

I'm just trying to figure out what place means. A "place" is an
equivalence class of functions. What does it even mean to evaluate a
number in a number field at one of the field's places then? I know how
to apply a single function to a number, but how do I apply an entire
equivalence class of functions to a number? It would be helpful to
figure out some of the other neat identities and results in these
papers about Weil height and figure out what the equivalent results
are for monzos.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I'm just trying to figure out what place means. A "place" is an
> equivalence class of functions. What does it even mean to evaluate a
> number in a number field at one of the field's places then?

It means you should use a canonical choice of function, which is what I was talking about.

If you want to fool around with this, I suggest downloading Pari, which lets you factor polynomials over p-adic numbers. If you take a monic minimal polynomial for an algebraic number, it will be irreducible by definition over Q, but may factor over Qp. The different monic factors correspond to different places lying over p, and you can evaluate each one in one canonical way by taking the norm term, finding the p-adic absolute value, and taking the nth root, where n is the degree of the factor in question.