Even of level n

Th recent discussion of generators for DE scales prompted me to ask
what a good generalization might be. It can't just come down to images
under some regular temperament of a Fokker block, since those will not
always have the property that the number of different scale step sizes
equals the rank of the temperament. Here's a proposed definition.

A DE scale is a scale which divides the octave by steps of two sizes,
such that any n contiguous steps (an interval class) comes in at most
two sizes. The minimal size of class of just one size is the period, and
a class with all but one of one size is a generator.

We define "even of level 1" to mean DE, and define even of level n
recursively, by requiring that if we amalgamate any two step sizes
among the set of n step sizes into a single step size, the result is
even of level n-1. Note that this definition does not require we know
the actual sizes, and hence does not depend on tuning, so I should
perhaps I should say step types, not step sizes.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> We define "even of level 1" to mean DE, and define even of level n
> recursively, by requiring that if we amalgamate any two step sizes
> among the set of n step sizes into a single step size, the result is
> even of level n-1.

Much too restrictive, since the amalgamated steps need to be
contiguous. But saying the result should be even of some level should
work.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > We define "even of level 1" to mean DE, and define even of level n
> > recursively, by requiring that if we amalgamate any two step sizes
> > among the set of n step sizes into a single step size, the result is
> > even of level n-1.
>
> Much too restrictive, since the amalgamated steps need to be
> contiguous. But saying the result should be even of some level should
> work.

It occurs to me that this is all the wrong approach. Must simpler and
better: a scale with n sizes of steps is n-even if every interval
class has at most n members. Then a DE scale is 2-even, and so forth.
An edo is 1-even.

>It occurs to me that this is all the wrong approach. Must simpler and
>better: a scale with n sizes of steps is n-even if every interval
>class has at most n members. Then a DE scale is 2-even, and so forth.
>An edo is 1-even.

That sounds good.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Th recent discussion of generators for DE scales prompted me to ask
> what a good generalization might be. It can't just come down to
images
> under some regular temperament of a Fokker block, since those will
not
> always have the property that the number of different scale step
sizes
> equals the rank of the temperament. Here's a proposed definition.
>
> A DE scale is a scale which divides the octave by steps of two
sizes,
> such that any n contiguous steps (an interval class) comes in at
most
> two sizes. The minimal size of class of just one size is the
period, and
> a class with all but one of one size is a generator.
>
> We define "even of level 1" to mean DE, and define even of level n
> recursively, by requiring that if we amalgamate any two step sizes
> among the set of n step sizes into a single step size, the result is
> even of level n-1. Note that this definition does not require we
know
> the actual sizes, and hence does not depend on tuning, so I should
> perhaps I should say step types, not step sizes.

We've seen this before on the tuning list. It was around one of the
few times a big-name academic posted there.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > We define "even of level 1" to mean DE, and define even of level n
> > recursively, by requiring that if we amalgamate any two step sizes
> > among the set of n step sizes into a single step size, the result is
> > even of level n-1.
>
> Much too restrictive, since the amalgamated steps need to be
> contiguous.

What do you mean, contiguous? And why?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > We define "even of level 1" to mean DE, and define even of
level n
> > > recursively, by requiring that if we amalgamate any two step
sizes
> > > among the set of n step sizes into a single step size, the
result is
> > > even of level n-1.
> >
> > Much too restrictive, since the amalgamated steps need to be
> > contiguous. But saying the result should be even of some level
should
> > work.
>
> It occurs to me that this is all the wrong approach. Must simpler
and
> better: a scale with n sizes of steps is n-even if every interval
> class has at most n members. Then a DE scale is 2-even, and so
forth.
> An edo is 1-even.

Are you sure this is different from your original, non-"contiguous"-
insisting formulation?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> We've seen this before on the tuning list. It was around one of the
> few times a big-name academic posted there.

Who was that, and what was the actual definition?

I've found that using what seems to be the nice definition for rank-n
evenness leads to a small number of qualifying scales. Especially for
larger scales, it seems to enforce strong regularity and smoothness.

Examples of rank-3 even scales are the famous zarlino scale and its
inverse, redfield, and the scale variously known as the Ramis monochord,
tamil_vi, schisdia1 and syndia6. I also found a few scales which only
turned up before (judging by the Scala archives) in the course of some
of my complete surveys. However, these sort of scales do not seem
common and an expeditious way of constructing them could be useful.

For an example of a tempered even rank-3 scale, tempering ramis via
64/63 planar surves the purpose, giving some tetrads to work with.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Are you sure this is different from your original, non-"contiguous"-
> insisting formulation?

No, but it's clearly a better approach.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Are you sure this is different from your original, non-"contiguous"-
> > insisting formulation?
>
> No, but it's clearly a better approach.

Maybe if it isn't different, it's not "better", it's merely an
alternative perspective on the same entity, and hence might be useful.