Tenny-reduced and Kees-reduced pitches

(moved here from the tuning list)

/tuning/topicId_64170.html#64599

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > > > > And as usual, I think it would make a lot more sense to
> > > > > reduce the pitches (thus making the choice of kernel basis
> > > > > irrelevant) instead of reducing the kernel basis and then
> > > > > constructing an FPB.
> > > >
> > > >
> > > > Ah yes, i could tell Tonescape to construct a 79-tone
> > > > periodicity-block first, *then* use 159-edo for the tuning.
> > >
> > > Not what I meant, but worth considering.
> >
> > Then can you please elaborate on what you did mean?
>
> Do the same kind of reduction you did to the kernel basis,
> only do it to the pitch ratios instead.
>
> > It seems i'm not understanding you well here.
>
> I tried to explain below.
>
> > > > Why do you say that reducing the number of pitches
> > > > makes "the choice of kernel basis irrelevant"?
> > >
> > > No, I meant Tenney-reducing the pitches (or better
> > > yet, Kees-reducing them), rather than Tenney-reducing
> > > the kernel basis.
> > >
> > > > The choice of kernel basis still determines how
> > > > compact the periodicity-block is.
> > >
> > > Not at all true, if you reduce the pitches. Then the
> > > choice of kernel basis becomes irrelevant -- any valid
> > > kernel basis yields the same final result. Which, BTW,
> > > is more compact than any Fokker periodicity block
> > > arising from any valid kernel basis.
> >
> >
> > OK, now i have to confess my ignorance. I've spent a
> > lot of time in the past year working on Tonescape and
> > only skimming past many posts on these lists. What does
> > it mean to Tenney- (or Kees-) reduce the pitches?
>
> One way of saying it is that it means to find the simplest
> ratio for each pitch that you can obtain by starting with
> the pitch's ratio in the Fokker periodicity block,
> periodicity strip, or periodicity sheet (whichever
> vanishing commas you use to define that) and transposing
> it by any number/combination of vanishing commas. The
> result will be the same no matter which Fokker periodicity
> block, strip, or sheet you start with. Gene does this a
> whole lot on the tuning-math list.
>
> > It seems that this can apply directly to how Tonescape
> > works,
>
> You better believe it!
>
> > so i definitely want to learn more.
>
> I suggest posting your questions on this to the tuning-math list.

So here i am.

Can someone please provide an example of this?
And please be as long-winded and detailed as possible.

Thanks.

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

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

> Can someone please provide an example of this?
> And please be as long-winded and detailed as possible.

Let's consider two examples of what number theorists call a height
function, Kees height and Tenney height. The Kees height of a positive
rational number is what results from removing all powers of two, so
that we have a ratio of odd integers, and then taking the maximum of
the numerator and denominator. The Tenney height, on the other hand,
is the product of the numerator and denominator. Hence the Kees height
of 10/9 would be 9, the maximum of 5 and 9, and the Tenny height would
be 90, the product of 10 and 9.

Suppose we have an et val, for instance <12 19 28|. For this we can
find a kernel basis, for instance {81/80, 128/125}. Now suppose we
have a 3 or 5 limit Fokker block; which therefore could just be the
Pythagoran 12-note scale, or the Ellis duodene, etc. Suppose for each
note q of this scale we consider the products q * (81/80)^i *
(128/125)^j, and pick the one with the smallest Kees height, breaking
any ties by picking the smallest Tenney height. We get the following
scale, which I once obtained previously as the "Hahn reduced" 5-limit
12-note scale, by the way:

! kesred12_5.scl
Kees reduced 5-limit 12-note scale = Hahn reduced
12
!
16/15
9/8
6/5
5/4
4/3
25/18
3/2
8/5
5/3
9/5
15/8
2

Note that the tie-breaker had to be invoked--9/8 and 10/9 both have a
Kees height of 9, as do 9/5 and 16/9.

Hi Gene,

Wow, that was painless ... a lot easier to understand
than i had expected. Thanks.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > Can someone please provide an example of this?
> > And please be as long-winded and detailed as possible.
>
> Let's consider two examples of what number theorists call a
> height function, Kees height and Tenney height. The Kees height
> of a positive rational number is what results from removing
> all powers of two, so that we have a ratio of odd integers,
> and then taking the maximum of the numerator and denominator.
> The Tenney height, on the other hand, is the product of the
> numerator and denominator. Hence the Kees height of 10/9
> would be 9, the maximum of 5 and 9, and the Tenny height
> would be 90, the product of 10 and 9.
>
> Suppose we have an et val, for instance <12 19 28|. For this
> we can find a kernel basis, for instance {81/80, 128/125}.
> Now suppose we have a 3 or 5 limit Fokker block; which
> therefore could just be the Pythagoran 12-note scale, or
> the Ellis duodene, etc. Suppose for each note q of this
> scale we consider the products q * (81/80)^i * (128/125)^j,
> and pick the one with the smallest Kees height, breaking
> any ties by picking the smallest Tenney height. We get the
> following scale, which I once obtained previously as the
> "Hahn reduced" 5-limit 12-note scale, by the way:
>
> ! kesred12_5.scl
> Kees reduced 5-limit 12-note scale = Hahn reduced
> 12
> !
> 16/15
> 9/8
> 6/5
> 5/4
> 4/3
> 25/18
> 3/2
> 8/5
> 5/3
> 9/5
> 15/8
> 2
>
> Note that the tie-breaker had to be invoked--9/8 and 10/9
> both have a Kees height of 9, as do 9/5 and 16/9.

>> Can someone please provide an example of this?
>> And please be as long-winded and detailed as possible.
>
>Let's consider two examples of what number theorists call a height
>function, Kees height and Tenney height. The Kees height of a positive
>rational number is what results from removing all powers of two, so
>that we have a ratio of odd integers, and then taking the maximum of
>the numerator and denominator. The Tenney height, on the other hand,
>is the product of the numerator and denominator. Hence the Kees height
>of 10/9 would be 9, the maximum of 5 and 9, and the Tenny height would
>be 90, the product of 10 and 9.
>
>Suppose we have an et val, for instance <12 19 28|. For this we can
>find a kernel basis, for instance {81/80, 128/125}. Now suppose we
>have a 3 or 5 limit Fokker block; which therefore could just be the
>Pythagoran 12-note scale, or the Ellis duodene, etc. Suppose for each
>note q of this scale we consider the products q * (81/80)^i *
>(128/125)^j, and pick the one with the smallest Kees height, breaking
>any ties by picking the smallest Tenney height. We get the following
>scale, which I once obtained previously as the "Hahn reduced" 5-limit
>12-note scale, by the way:
>
>! kesred12_5.scl
>Kees reduced 5-limit 12-note scale = Hahn reduced
>12
>!
>16/15
>9/8
>6/5
>5/4
>4/3
>25/18
>3/2
>8/5
>5/3
>9/5
>15/8
>2
>
>Note that the tie-breaker had to be invoked--9/8 and 10/9 both have a
>Kees height of 9, as do 9/5 and 16/9.

Great explanation, Gene. That could be cut and pasted right into
a FAQ.

-Carl

Gene, the way you're breaking ties here ruins the compactness of the block. 25/18 isn't consonant with anything in your block, but wiuld be with 10/9. I think I have a better suggestion. In case of a tie, re-state the ratios as intervals from 3/2 instead of 1/1, then use Kees expressibility instead. Tenney HD isn't really appropriate here because the scales go down below 1/1 as well as up above it -- these are octave-repeating scales Monz is dealing with.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > Can someone please provide an example of this?
> > And please be as long-winded and detailed as possible.
>
> Let's consider two examples of what number theorists call a height
> function, Kees height and Tenney height. The Kees height of a positive
> rational number is what results from removing all powers of two, so
> that we have a ratio of odd integers, and then taking the maximum of
> the numerator and denominator.

If we call the result K, then this is the same as saying the ratio is a "ratio of K" in Partch's terminology. In other words, this is what we colloquially call the "odd limit" of the ratio.

> The Tenney height, on the other hand,
> is the product of the numerator and denominator. Hence the Kees height
> of 10/9 would be 9, the maximum of 5 and 9, and the Tenny height would
> be 90, the product of 10 and 9.
>
> Suppose we have an et val, for instance <12 19 28|. For this we can
> find a kernel basis, for instance {81/80, 128/125}. Now suppose we
> have a 3 or 5 limit Fokker block; which therefore could just be the
> Pythagoran 12-note scale, or the Ellis duodene, etc. Suppose for each
> note q of this scale we consider the products q * (81/80)^i *
> (128/125)^j, and pick the one with the smallest Kees height, breaking
> any ties by picking the smallest Tenney height. We get the following
> scale, which I once obtained previously as the "Hahn reduced" 5-limit
> 12-note scale, by the way:
>
> ! kesred12_5.scl
> Kees reduced 5-limit 12-note scale = Hahn reduced
> 12
> !
> 16/15
> 9/8
> 6/5
> 5/4
> 4/3
> 25/18
> 3/2
> 8/5
> 5/3
> 9/5
> 15/8
> 2
>
> Note that the tie-breaker had to be invoked--9/8 and 10/9 both have a
> Kees height of 9, as do 9/5 and 16/9.
>
I have no idea why you needed to replace the complexity measures with their exponentials -- being a number-theoretic "height" function (what are the criteria for that?) doesn't seem to make the reduction any easier.

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

> Gene, the way you're breaking ties here ruins the compactness of the
block. 25/18 isn't consonant with anything in your block, but wiuld be
with 10/9. I think I have a better suggestion. In case of a tie,
re-state the ratios as intervals from 3/2 instead of 1/1, then use
Kees expressibility instead. Tenney HD isn't really appropriate here
because the scales go down below 1/1 as well as up above it -- these
are octave-repeating scales Monz is dealing with.

I'll try that. Another suggestion would be to use Tenney height from
1/sqrt(2) to sqrt(2).

> I have no idea why you needed to replace the complexity measures
with their exponentials -- being a number-theoretic "height" function
(what are the criteria for that?) doesn't seem to make the reduction
any easier.

Number theorists often use logarithmic height functions, and I don't
have to do it non-logarithmically. However, if I don't I'm simply
working with integer-valued functions, which is a lot nicer in some
ways. I certainly prefer it when programming something.

Why is it a big deal?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@...>
wrote:
>
> Gene, the way you're breaking ties here ruins the compactness of the
block. 25/18 isn't consonant with anything in your block, but wiuld be
with 10/9.

Tie breaking isn't the problem; the only possibilities with a Kees
height under 100 are 25/18, 36/25, 45/32, and 64/45, and the last two
are simply worse from either a Kees or Tenney point of view. At least
you get a diminished seventh chord.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@>
> wrote:
> >
> > Gene, the way you're breaking ties here ruins the compactness of the
> block. 25/18 isn't consonant with anything in your block, but wiuld be
> with 10/9.
>
> Tie breaking isn't the problem;

Yes it is.

> the only possibilities with a Kees
> height under 100 are 25/18, 36/25,

These two both have a "Kees height" (since you're re-engineering terminology) of 25, Gene, so it's a tie. If you use the rule I suggested, you break it in favor of 36/25, which is going to form a consonance with 9/5, etc.

> 45/32, and 64/45, and the last two
> are simply worse from either a Kees or Tenney point of view.

So why did you bother mentioning them?

> At least
> you get a diminished seventh chord.
>
Please think for real about what I'm saying.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@>
> wrote:
>
> > Gene, the way you're breaking ties here ruins the compactness of the
> block. 25/18 isn't consonant with anything in your block, but wiuld be
> with 10/9. I think I have a better suggestion. In case of a tie,
> re-state the ratios as intervals from 3/2 instead of 1/1, then use
> Kees expressibility instead. Tenney HD isn't really appropriate here
> because the scales go down below 1/1 as well as up above it -- these
> are octave-repeating scales Monz is dealing with.
>
> I'll try that. Another suggestion would be to use Tenney height from
> 1/sqrt(2) to sqrt(2).

Nah.

> > I have no idea why you needed to replace the complexity measures
> with their exponentials -- being a number-theoretic "height" function
> (what are the criteria for that?) doesn't seem to make the reduction
> any easier.
>
> Number theorists often use logarithmic height functions, and I don't
> have to do it non-logarithmically. However, if I don't I'm simply
> working with integer-valued functions, which is a lot nicer in some
> ways. I certainly prefer it when programming something.
>
> Why is it a big deal?
>
Just trying to avoid an overabundance of definitions, lingo, etc. Why do with two entities what you can do with one?

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

> Please think for real about what I'm saying.

So is the rule that (3/2)/(25/18) = 27/25, and (3/2)/(36/25) = 25/24,
so that we choose 36/25?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@>
> > wrote:
> > >
> > > Gene, the way you're breaking ties here ruins the compactness of the
> > block. 25/18 isn't consonant with anything in your block, but wiuld be
> > with 10/9.
> >
> > Tie breaking isn't the problem;
>
> Yes it is.
>
> > the only possibilities with a Kees
> > height under 100 are 25/18, 36/25,
>
> These two both have a "Kees height" (since you're re-engineering terminology) of 25, Gene, so it's a tie. If you use the rule I suggested, you break it in favor of 36/25, which is going to form a consonance with 9/5, etc.

By "etc.", I mean 6/5 too. 25/18 was only consonant with one note, 5/3.
>
> > 45/32, and 64/45, and the last two
> > are simply worse from either a Kees or Tenney point of view.
>
> So why did you bother mentioning them?
>
> > At least
> > you get a diminished seventh chord.
> >
> Please think for real about what I'm saying.
>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@>
> wrote:
>
> > Please think for real about what I'm saying.
>
> So is the rule that (3/2)/(25/18) = 27/25, and (3/2)/(36/25) = 25/24,
> so that we choose 36/25?

Right. It's just an example of the kind of rule we might use. More sophisticated, but a bit less accessible, is to make a norm or distance measure corresponding to Kees expressibility (which we've already done) on the lattice, pick a point on the lattice that's not on a note or between two notes, and reduce each note by minimizing its "Kees distance" from that point, Then no ties will arise in the first place.

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

> Just trying to avoid an overabundance of definitions, lingo, etc.
Why do with two entities what you can do with one?

Just to add to the overabundence, max(numerator, denominator) or
equivalently log(max(numerator, denominator)) is sometimes called the
Diophantine height or Weil height. "Bit height" can be used for Tenney
height or whatever you call it, since log2(numerator*denominator)
measures about how many bits of data storage you need; it's more
associated with computational questions. Also many more much more
complicated things than the positive rationals have heights, but we
needn't worry about them.