maximal evenness

Paul,

The first I ever heard of maximal evenness (and the only time other
than the occasional reference here at the tuning list) was the
unflattering bit in your 22-tET paper. But I don't think that or any
of the reasons you keep offering for not liking it (or rather for not
thinking it's important/useful in some way) address any of the reasons
why I think it is.

I guess the most important of those would be the relation to the
Moment of Symmetry... if I hadn't of seen the two term M-out-of-N
connection there I probably never would've seen the three term
L-out-of-M-out-of-N scales. And seeing this connection helped me solve
a very difficult problem.

No maximal evenness then no seeing single generator interpretations
for fractional periodicity. No maximal evenness and no seeing a global
ordering method for the steps of N-bonacci type scales... (etceteras)

In short, I'm glad I didn't take your word for it on ME!

--Dan Stearns

someone please write a good definition of
"maximal evenness" for me. thanks.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> someone please write a good definition of
> "maximal evenness" for me. thanks.

Isn't that what Blackwood is getting at in his chapters about spirals
of infinite meantone, or infinite Pythagorean? The choice of a
consistent fifth-size within such a spiraling system generates tones
and semitones that (if we keep all their functions properly separate,
as to chromatic vs diatonic) are equal in size to one another, and
therefore melodies are smooth.

Unless you were referring to something entirely different. :)

Brad Lehman

hi Brad,

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > someone please write a good definition of
> > "maximal evenness" for me. thanks.
>
> Isn't that what Blackwood is getting at in his chapters
> about spirals of infinite meantone, or infinite Pythagorean?
> The choice of a consistent fifth-size within such a spiraling
> system generates tones and semitones that (if we keep all
> their functions properly separate, as to chromatic vs diatonic)
> are equal in size to one another, and therefore melodies
> are smooth.
>
> Unless you were referring to something entirely different. :)

i never really knew exactly what was meant by
"maximal evenness", but it is a term that has been
used by tuning theorists and it needs to have a
definition in my Encyclopaedia.

i'm not sure how it intersects with what Blackwood says.
but i have his book ... please cite the page numbers
and i'll take a look!

-monz

I've written this in the Scala tips.par file. Feel free to use it or
clarify it further.

A scale is maximally even if there are no more than two different interval
sizes for each interval class (this can be seen with SHOW INTERVALS), and
if
the difference between each size of interval per interval class is less
than
one scale step. Originally, this one scale step was intended to be the step
of
some equal tempered scale from which a subset was taken.
Both versions of this property are shown by SHOW DATA. If the scale is a
subset of an equal temperament with less than 1200 tones and maximally
even,
it's said that it's ME for that particular equal temperament. For the more
general version, it's said that it's ME for L / S <= 2, the ratio of the
size of larger step (step being interval of class 1) to the size of the
smallest step must be less than or equal to two. See also FIT/MODE.

A weaker form, distributional evenness, is also mentioned by SHOW DATA. For
this the scale must have only one or two different interval sizes for each
interval class. So if a scale is maximal even, it is also distributional
even.
The command FIT/MODE shows whether these properties exist for an equal
temperament mode. For maximal evenness, the definition here is in one
respect
different than with SHOW DATA: the difference in step sizes may not be
greater
than 1. This is the original definition of John Clough. For distributional
evenness, Clough's definition is not restricted to integer step size
proportions, so there is no difference of definition there.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul" <manuel.op.de.
coul@e...> wrote:
>
> I've written this in the Scala tips.par file. Feel free to use it or
> clarify it further.
>
>
> A scale is maximally even if there are no more than
> two different interval sizes for each interval class
> (this can be seen with SHOW INTERVALS), and if
> the difference between each size of interval per
> interval class is less than one scale step.
> <etc. -- snip>

thanks, Manuel.

i've split up what you wrote in that post and put part
of it into "maximal evenness" and the other part into
"distributional evenness".

Paul also gave me a definition offlist.

unfortunately, right now my server seems to be taking
a nap. :( i'll upload the new pages as soon as it
wakes up.

-monz

--- In tuning@yahoogroups.com, "Manuel Op de Coul" <manuel.op.de.
coul@e...> wrote:
> A scale is maximally even if there are no more than two different
interval
> sizes for each interval class (this can be seen with SHOW INTERVALS)
, and
> if
> the difference between each size of interval per interval class is
less
> than
> one scale step.

Makes sense.

But, how much is "maximal evenness" really a desirable musical
virtue?
(There's a big aesthetics question.)

I was studying a piece today that makes especially expressive use of
four different sizes of semitones, all of them 100 cents or smaller;
it sounds like nothing so special if the collection of semitones is
sanded down to be only two or one available size(s). The piece
quickly loses character (and, for me, my interest as a listener) if
all these subtly different expressive moments are reduced to sameness.

Brad Lehman

On Tuesday 31 August 2004 05:12 pm, Brad Lehman wrote
> But, how much is "maximal evenness" really a desirable musical
> virtue?
> (There's a big aesthetics question.)
>
> I was studying a piece today that makes especially expressive use of
> four different sizes of semitones, all of them 100 cents or smaller;
> it sounds like nothing so special if the collection of semitones is
> sanded down to be only two or one available size(s). The piece
> quickly loses character (and, for me, my interest as a listener) if
> all these subtly different expressive moments are reduced to sameness.

By all means, share the title and composer with us of said piece !!!

Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

> > I was studying a piece today that makes especially expressive use
of
> > four different sizes of semitones, all of them 100 cents or
smaller;
> > it sounds like nothing so special if the collection of semitones
is
> > sanded down to be only two or one available size(s). The piece
> > quickly loses character (and, for me, my interest as a listener)
if
> > all these subtly different expressive moments are reduced to
sameness.
>
> By all means, share the title and composer with us of said piece !!!

It's described fully in an article I've written for a journal. Until
its publication in 2005 I'm not at liberty to provide details, except
to assert that the piece is by a composer who had an extraordinarily
keen ear for intonation, and who knew exactly what he was doing with
his resources. And, it's from long before the invention of "cents"
measurement.

If you want to hear and see some really exciting and unsubtle uses of
tiny semitones as special effect, check out Don Carlo Gesualdo's
keyboard piece, _Canzon francese_, if played in any regular (i.e.
meantone) temperament, or in an Italian or French irregular
temperament. The chromatic trills in there, written-out with correct
spelling in the score, are hair-raising.

And, there are plenty of examples in Marais' music for viola da gamba
where the player is to glide from the third of a minor chord to the
third of a major chord, i.e. with a chromatic (color-changing)
semitone, above a steady bass: an unforgettable effect.

Brad Lehman

From Tonalsoft Encyclopedia:

maximal evenness

[from Paul Erlich, private communication]

Examples of maximally even scales in a universe of 12 pcs include 2-
1-2-1-2-1-2-1 and 2-2-1-2-2-2-1, because they are, in a particular
sense, the best approximations of 8-equal and 7-equal available in
12-equal.

The diatonic scale in 31-equal is not maximally even, which in my
opinion, shows how un-useful the maximal evenness concept is :) The
maximally even 7-note scale in 31-equal is 4-4-5-4-5-4-5, not the
diatonic 5-5-3-5-5-5-3.

[me]

But if you consider the diatonic set as maximally even in a
chromatic set and this one as maximally even in a 31-equal you get
exactly the diatonic 5535553, which in my opinion shows that this
concept works.

Lorenzo

What about our 79/80 being a maximally even MOS of 159-equal?

Oz.

----- Original Message -----
From: "lorenzofrizzera" <lorenzo.frizzera@cdmrovereto.it>
To: <tuning@yahoogroups.com>
Sent: 15 Ekim 2006 Pazar 1:31
Subject: [tuning] maximal evenness

> >From Tonalsoft Encyclopedia:
>
> maximal evenness
>
> [from Paul Erlich, private communication]
>
> Examples of maximally even scales in a universe of 12 pcs include 2-
> 1-2-1-2-1-2-1 and 2-2-1-2-2-2-1, because they are, in a particular
> sense, the best approximations of 8-equal and 7-equal available in
> 12-equal.
>
> The diatonic scale in 31-equal is not maximally even, which in my
> opinion, shows how un-useful the maximal evenness concept is :) The
> maximally even 7-note scale in 31-equal is 4-4-5-4-5-4-5, not the
> diatonic 5-5-3-5-5-5-3.
>
> [me]
>
> But if you consider the diatonic set as maximally even in a
> chromatic set and this one as maximally even in a 31-equal you get
> exactly the diatonic 5535553, which in my opinion shows that this
> concept works.
>
> Lorenzo
>

--- In tuning@yahoogroups.com, "lorenzofrizzera"
<lorenzo.frizzera@...> wrote:

> The diatonic scale in 31-equal is not maximally even, which in my
> opinion, shows how un-useful the maximal evenness concept is :)

They have more than two intervals per interval class, which I think
means they are maximally even.

the proportion of large in to small seems like it would be better when clearly defined .
for instance the golden proportion (1.618..../1) might be relation

maximal evenness seems like a good method to create ambiguousness.
look at the whole tone scale, or the diminished scale.
good to have a little ambiguousness periodically though

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles