Chord centered Kees reduction

Paul Erlich suggested Kees reduction using a center not a lattice
point. Below I reduce the 5-1imit, 12-note scale with Kees reduction
with respect to the major triad centroid, |* 1/3 1/3>. This is
centrally located in the symmetrical note-class metric, but is skewed
in the Kees metric so that ties are automatically broken. I get the
following:

! kred12_5.scl
Kees reduced 5-limit centered on |1 1 1>/3 = rousseau.scl
12
!
25/24
9/8
6/5
5/4
4/3
25/18
3/2
8/5
5/3
9/5
15/8
2

The Rousseau scale is "Rousseau's Monochord, Dictionnaire de musique
(1768)" according to the Scala file. The inverse scale is marpurg4,
which is "Marpurg 4, also Yamaha Pure Minor" according to Scala. It's
clearly an improvement on the other Kees reduction.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> Paul Erlich suggested Kees reduction using a center not a lattice
> point. Below I reduce the 5-1imit, 12-note scale with Kees reduction
> with respect to the major triad centroid, |* 1/3 1/3>. This is
> centrally located in the symmetrical note-class metric, but is
skewed
> in the Kees metric so that ties are automatically broken. I get the
> following:
>
> ! kred12_5.scl
> Kees reduced 5-limit centered on |1 1 1>/3 = rousseau.scl

I think you mean |* 1 1>/3 . . .

> 12
> !
> 25/24
> 9/8
> 6/5
> 5/4
> 4/3
> 25/18
> 3/2
> 8/5
> 5/3
> 9/5
> 15/8
> 2

25/18-25/24
...\.../.\
....\./...\
....5/3---5/4--15/8
..../.\.../.\.../.\
.../...\./...\./...\
.4/3---1/1---3/2---9/8
..\..../.\.../.\.../
...\../...\./...\./
....8/5---6/5---9/5

Not the most compact arrangement I've seen, but this approach might
be great for certain cardinalities (15? 19?) . . .

> The Rousseau scale is "Rousseau's Monochord, Dictionnaire de musique
> (1768)" according to the Scala file. The inverse scale is marpurg4,
> which is "Marpurg 4, also Yamaha Pure Minor" according to Scala.
It's
> clearly an improvement on the other Kees reduction.

I don't know if it's the best way, though . . .

Another idea, which I think will prevent ties, is to use the center
point [* epsilon 0 0 0 0 . . . 0>, where you take the limit as
epsilon approaches zero from the right. This will be more centered
around 1/1. And it's not as "triad-centric" . . .

I hope Monz will be able to implement one of these more sophisticated
approaches, or at least the tie-breaking rule I proposed, when he
adds this kind of reduction/rationalization feature to Tonescape.

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

> > ! kred12_5.scl
> > Kees reduced 5-limit centered on |1 1 1>/3 = rousseau.scl
>
> I think you mean |* 1 1>/3 . . .

Doesn't matter.

> Not the most compact arrangement I've seen, but this approach might
> be great for certain cardinalities (15? 19?) . . .

It's interesting from the point of view of a starling temperament
target, where 25/18 ~ 7/5 and 25/24 ~ 21/20.

> Another idea, which I think will prevent ties, is to use the center
> point [* epsilon 0 0 0 0 . . . 0>, where you take the limit as
> epsilon approaches zero from the right. This will be more centered
> around 1/1. And it's not as "triad-centric" . . .

Epsilon could just be a positive infinitesimal. But why is this better?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@>
> wrote:
>
> > > ! kred12_5.scl
> > > Kees reduced 5-limit centered on |1 1 1>/3 = rousseau.scl
> >
> > I think you mean |* 1 1>/3 . . .
>
> Doesn't matter.
>
> > Not the most compact arrangement I've seen, but this approach might
> > be great for certain cardinalities (15? 19?) . . .
>
> It's interesting from the point of view of a starling temperament
> target, where 25/18 ~ 7/5 and 25/24 ~ 21/20.
>
> > Another idea, which I think will prevent ties, is to use the center
> > point [* epsilon 0 0 0 0 . . . 0>, where you take the limit as
> > epsilon approaches zero from the right. This will be more centered
> > around 1/1. And it's not as "triad-centric" . . .
>
> Epsilon could just be a positive infinitesimal. But why is this better?
>
(a) it comes a lot closer to Monz's goal of symmetry around 1/1

(b) triads are not special in the Kees approach, even in the 5-limit

(c) the central chord will be missing composite identities in the 11-limit or higher

Would it be too hard to implement my suggestion for a few cardinalities and limits (or just 12-note 5-limit) so that we may see what it gives and compare to other methods?

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

> Would it be too hard to implement my suggestion for a few
cardinalities and limits (or just 12-note 5-limit) so that we may see
what it gives and compare to other methods?
>

I thought about how to do this, and concluded it was equivalent to the
following proceedure: choose the least Kees height. If two such
heights are equal, then choose the one where the denominator of the
odd part is least.

This seems like a reasonable proceedure, but infortunately in the
5-limit for 12 notes it gives exactly the scale we had before from
using Tenney height to break ties. What do you think?