Stretched EDOs and octave consistency

If you are working with an EDO (Equal division of the Octave)
and want to stretch or compress it AND maintain consistency
at the octave (which will maintan all other consistencys as
well), then the way I've been thinking about it is...

max_stretch := 1 + (1/2)*(1/N)*(1/R)
max_comress := 1 - (1/2)*(1/N)*(1/R)

where N is the number of the EDO and
R is the number of octaves you care about being
consistent in

So, for instance, a maximally stretched 19 tet across 4
octaves would have

max_stretch := 1.006578947

The tones in the scale are given by

0 2^(0*max_stretch/19)
1 2^(1*max_stretch/19)
...
4*19

and if you convert the whole thing to cents you find that
this last scale degree is 32 cents sharp, which is half a
'chroma' in 19, the maximum error which maintains
consistency.

This may be a bit pedestrian for the kind of warping that
Jacky and Dan are discussing, but its something I'd wanted
to figure out.

Bob Valentine

Robert C Valentine wrote,

<<This may be a bit pedestrian for the kind of warping that Jacky and
Dan are discussing,>>

Hi Robert,

I think this is very interesting. However I'm not sure it does exactly
what you seem to be saying it does... 17-tET for instance is only
consistent through the 3-limit, but if you use the method that you
gave and let R=3 then N, or ~16.835-tET, becomes consistent through to
the 5-limit.

If this type of non-octave consistency is what you meant by
"consistency" then I think the results of the stretch (or compression)
does effect the overall consistency of a given N.

--Dan Stearns

Dan Stearns wrote:

> I think this is very interesting. However I'm not sure it does exactly
> what you seem to be saying it does... 17-tET for instance is only
> consistent through the 3-limit, but if you use the method that you
> gave and let R=3 then N, or ~16.835-tET, becomes consistent through to
> the 5-limit.

What Robert wrote seems to be correct for consistency between the
stretched and unstretched scale, but not between the stretched scale and
any set of JI intervals.

> If this type of non-octave consistency is what you meant by
> "consistency" then I think the results of the stretch (or compression)
> does effect the overall consistency of a given N.

With an odd-limit, the consistency can only go down if you stretch (+/-)
the octaves. And the best tuning for an odd-limit must always have just
octaves. Really, for non-octave scales you need to consider an
octave-specific set of intervals. So include fifths but not fourths.
It's not quite that simple, because you still need both kinds of thirds,
but if you thought about it I'm sure you could come up with something.
Indeed, the Bohlen-Pierce and 88CET scales are a radical take on this
approach.

Graham

--- In tuning@y..., graham@m... wrote:

Really, for non-octave scales you need to consider an
> octave-specific set of intervals.

This is really what I was attempting to do with my initial post,
using a 5 limit JI scale as the object of contortion.

Jacky Ligon

Graham Breed wrote,

<<With an odd-limit, the consistency can only go down if you stretch
(+/-) the octaves. And the best tuning for an odd-limit must always
have just octaves. Really, for non-octave scales you need to consider
an octave-specific set of intervals.>>

By using an integer-limit (as opposed to the odd-limit) the overall
compass of what's consistent can go up. With 17-tET a slight stretch,
say 16.96-tET for example, nudges 5 into the overall consistency
scheme.

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Graham Breed wrote,
>
> <<With an odd-limit, the consistency can only go down if you stretch
> (+/-) the octaves. And the best tuning for an odd-limit must always
> have just octaves. Really, for non-octave scales you need to
consider
> an octave-specific set of intervals.>>
>
> By using an integer-limit (as opposed to the odd-limit) the overall
> compass of what's consistent can go up. With 17-tET a slight
stretch,
> say 16.96-tET for example, nudges 5 into the overall consistency
> scheme.
>
> --Dan Stearns

You know - this weekend I was thinking (yes, I do this on occasion);
if someone on the list can make the ultimate transposition of the
stretch logic to any limit consistency, then one could control the
limit flavors of the stretch. Well, I see Dan, Bob and Graham have
already planted their microtonal flags on this hilltop. Now I will
set back and learn.

Please post some examples and methods of stretch/non-octave tunings,
which align to given limits. You'll have my ED attention.

For me, the ultimate challenge is to find ways of doing what a
Gamelan type of tuning can. My crude Fibo-Stretch example was a
humble attempt in this direction.

The most interesting thing about it, is how consonant the scale
sounded (I played it for hours), which I didn't really expect. There
was this kind of aural bonding effect like that of JI - but
different. It just made sense as sounds overlapped - it was self-
supporting; creating its own little world and logic.

The idea of using a "curved" rather than linear stretch may allow one
to do some really fancy things with consistencies, but I just haven't
taken it there yet. This is the ultimate level for this concept; to
consciously chose which pitches to target.

Also as mentioned and shown in past articles, one can use a JI scheme
to achieve essentially the same thing, but a generalized method would
be of great use in the attempt to contort a scale to match certain
rational limits. ETs do seem like a good possible model as a template
scale, to which the curve may be applied, in order to "bend" the
scale toward more pleasing, or desired parameters.

Still what compels me about it, is to think of ways to tune some
beats into a scale, with some controlled method, which will yield a
scale that has aural properties like that of a Gamelan scale. I'm not
so sure that one will get this with a pure JI method, but then again,
my 23 limit non-octave scales (especially the utonal), has some of
these qualities to me. All are valid and useful, and I'd like to
understand both.

Thanks for the inspiration,

Jacky Ligon

Food for thought: Couldn't one find what might be "limit
trajectories" to align a scale to?

Jacky Ligon wrote:

> You know - this weekend I was thinking (yes, I do this on occasion);
> if someone on the list can make the ultimate transposition of the
> stretch logic to any limit consistency, then one could control the
> limit flavors of the stretch. Well, I see Dan, Bob and Graham have
> already planted their microtonal flags on this hilltop. Now I will
> set back and learn.

I haven't been following this thread as closely as I should. Have you
seen this list by Manuel and your old friend Paul Erlich yet?

<ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt>

Integer limits look like good general things to check for consistency.
But for optimizing I'm not so sure. It seems right to me that fifths
are more important than fourths, and major than minor thirds. Perhaps
if we optimized for prime ratios? A lot of the problems with that in
the octave-specific case magically fall away. That is, if fifths and
major thirds are both equally sharp, minor thirds will be just. But
if you shrink the octave, the major and minor thirds will improve, and
minor thirds won't be that bad. It sort of takes care of itself.

Graham

--- In tuning@y..., ligonj@n... wrote:

/tuning/topicId_20427.html#20451

> The idea of using a "curved" rather than linear stretch may
> allow one to do some really fancy things with consistencies,
> but I just haven't taken it there yet. This is the ultimate
> level for this concept; to consciously chose which pitches
> to target.

Wow - now *this* is some interesting stuff!

As far as I can envision it tho, you'd have to be talking
about some *REALLY* fancy things, because the whole point
of consistency concerns the linearities of musical notation.
Give the tuning a nonlinearity, and it would *seem* that
consistency goes right out the window.

Just for the benefit of you or anyone else who may have
missed it, the best exposition I've seen on consistency
(other than the numerous posts here by Pauls Erlich and Hahn)
is in Patrick Ozzard-Low's book, which can be downloaded
in its entirety from:
http://www.c21-orch-instrs.demon.co.uk/

(BTW, anyone heard from Patrick lately? I met him when he
passed thru San Diego a couple of years ago, but haven't
heard anything from him in a long time... I believe he
never returned to the list after his long trip.)

> Still what compels me about it, is to think of ways to tune
> some beats into a scale, with some controlled method, which
> will yield a scale that has aural properties like that of a
> Gamelan scale. I'm not so sure that one will get this with
> a pure JI method, but then again, my 23 limit non-octave scales
> (especially the utonal), has some of these qualities to me.
> All are valid and useful, and I'd like to understand both.

I don't wish to open up that "define just-intonation" can
of worms again, but depending on exactly what you mean by
JI, sure, I think you can acheive *anything* with JI.

It all depends on your choice of intervals/ primes/ odds/ etc.
But JI can be so flexible, if you extend the definition
correspondingly, that anything is possible. That's why I
love it so much.

It's certainly true, as was demonstrated with the Hammond
Organ rational tuning, that ratios can be so arbitrarily close
to an ET that there's no audible difference, but if the
tuning is rational, then it *is* rational.

Jacky, you really got me intrigued with that consistency
business...

-monz
http://www.monz.org
"All roads lead to n^0"

Jacky Ligon wrote:

> The idea of using a "curved" rather than linear stretch may allow one
> to do some really fancy things with consistencies, but I just haven't
> taken it there yet. This is the ultimate level for this concept; to
> consciously chose which pitches to target.
>
> Also as mentioned and shown in past articles, one can use a JI scheme
> to achieve essentially the same thing, but a generalized method would
> be of great use in the attempt to contort a scale to match certain
> rational limits. ETs do seem like a good possible model as a template
> scale, to which the curve may be applied, in order to "bend" the
> scale toward more pleasing, or desired parameters.

You know, there's a sense in which a harmonic fragment scale is an ET stretched on a curve - and
vice versa.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

David J. Finnamore wrote,

<<You know, there's a sense in which a harmonic fragment scale is an
ET stretched on a curve - and vice versa.>>

Hi David,

I agree, and I posted quite a bit along these lines a while back where
the basic idea was that you could mathematically morph an undertone
series into an overtone series, and that this morph would pass through
an near-equal area that could (in an analogous sense to the over and
under series) be seen as an "equaltone" series.

Here's a quick recap should anyone be interested and not already seen
it...

Where "n" is the cardinality (or distinct number of notes in a series)
and "x" is any given number, a series is defined as n*x with a
sequential numerator rule of +(x-2) and a sequential denominator rule
of -1. In other words, say n = 3 and x = 5. Then n*x = 15 and x-2 = 3,
and the sequence is 15/15, 18/14, 21/13, 24/12:

1/1 9/7 21/13 2/1

Or, in a more general sense:

n n+(x-2)
---, --------, ..., 1:2
n n-1

Letting x = 2 gives the corresponding n-undertone series, and
incrementally increasing the value of "x" by rationals works "n"
towards ever more accurate approximations of its corresponding
overtone series.

The interesting point of convergence occurs when x = sqrt(2)+2.

The simplest rational or RI interpretation of this is x = 3.5, and I
refer to this as the equaltone series. Amazingly, when x = sqrt(2)+2
the maximum difference between any n-tET interval and its
corresponding n-series interval is only ~2 cents.

--Dan Stearns

--- In tuning@y..., "David J. Finnamore" <daeron@b...> wrote:
> You know, there's a sense in which a harmonic fragment scale is an
ET stretched on a curve - and
> vice versa.

David, Kraig and Dan,

This is one of the methods I was refering to when I was implying that
this could be done with ratios. Perhaps I'll post a scale of this
nature. I have one really nice scale like David mentions here, that
I've created during this fascinating thread. Great thanks to all
participants!

Microtonally yours,

Jacky Ligon