Optimal Tuning Systems

Hi,

Some readers here may be interested in the writing and software concerning "Optimal Tuning Systems" by Larry Polansky et al.There was an ICMC paper just published and a "Perspectives of New Music" paper is submitted

http://eamusic.dartmouth.edu/~larry/owt/

Best
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

On Sep 6, 2008, at 10:31 AM, Torsten Anders wrote:
> Some readers here may be interested in the writing and software > concerning "Optimal Tuning Systems" by Larry Polansky et al.There > was an ICMC paper just published and a "Perspectives of New Music" > paper is submitted
> http://eamusic.dartmouth.edu/~larry/owt/
>

Just to make it a bit more specific: this is about a math. model and software which makes the construction of well-temperaments relatively strait-forward.

Torsten

>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

I assume this is the paper that came out of Sante Fe recently.
The title is a bit misleading, as their method dealt only
with WTs. Also, similar but superior (IMO) WT assessments
have been done on this list on several occasions.

-Carl

> Hi,
>
> Some readers here may be interested in the writing and software
> concerning "Optimal Tuning Systems" by Larry Polansky et al.
> There was an ICMC paper just published and a "Perspectives of
> New Music" paper is submitted
>
> http://eamusic.dartmouth.edu/~larry/owt/
>
> Best
> Torsten
>
> --
> Torsten Anders
> Interdisciplinary Centre for Computer Music Research
> University of Plymouth
> Office: +44-1752-586219
> Private: +44-1752-558917
> http://strasheela.sourceforge.net
> http://www.torsten-anders.de
>

Dear Carl,

What I liked about this approach is that your can rate the importance of intervals and keys and then the model computes you a well-temperament. If there are similar or even superior approaches to this, could you perhaps share some pointers to literature or software.

Thank you!

Best
Torsten

On Sep 6, 2008, at 5:46 PM, Carl Lumma wrote:
> I assume this is the paper that came out of Sante Fe recently.
> The title is a bit misleading, as their method dealt only
> with WTs. Also, similar but superior (IMO) WT assessments
> have been done on this list on several occasions.
>
> -Carl
>
> > Hi,
> >
> > Some readers here may be interested in the writing and software
> > concerning "Optimal Tuning Systems" by Larry Polansky et al.
> > There was an ICMC paper just published and a "Perspectives of
> > New Music" paper is submitted
> >
> > http://eamusic.dartmouth.edu/~larry/owt/
> >
> > Best
> > Torsten
> >
> > --
> > Torsten Anders
> > Interdisciplinary Centre for Computer Music Research
> > University of Plymouth
> > Office: +44-1752-586219
> > Private: +44-1752-558917
> > http://strasheela.sourceforge.net
> > http://www.torsten-anders.de
> >
>
>
>
--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

> Dear Carl,
>
> What I liked about this approach is that your can rate the
> importance of intervals and keys and then the model computes
> you a well-temperament. If there are similar or even superior
> approaches to this, could you perhaps share some pointers to
> literature or software.
>
> Thank you!
>
> Best
> Torsten

John deLaubenfels' software could minimize the squared
error over an arbitrary list of intervals *for a given
piece of music* (MIDI file). These are the so-called
COFT (Calculated Optimum Fixed Tuning).

I showed WTs with tempered octaves where, for certain
error functions such as unweighted squared error, the
average error over all keys of the WT is less than the
error of a key in 12-ET. Under other error functions
(such as Tenney-weighted error), these same WTs have a
higher average error than 12-ET. All this was actually
a trivial extension of the behavior of those error
functions on single chords.

I also presented evidence that when harmonic entropy is
the error function, no WT has an average key entropy as
low as the entropy of a key in 12-ET.

Next, I combinatorially explored WTs with 2 sizes of
5ths and pure octaves, which are easy to tune by ear.
In this class of WTs, only the 1/5- and 1/6-comma
varieties are sufficiently different from 12-ET on one
hand, and from 1/4-comma meantone on the other, to be
very interesting. I used Scheme programs to find all
1/5- and 1/6-comma WTs that meet certain constraints
(e.g. major 3rds not to exceed 404 cents).

But generally there's no magic bullet with WTs - there's
just not much room for magic. The idea that the ratio
between the beat rates of different intervals should be
simple was explored in great depth on this list and on
tuning-math. Many WTs with simple "brats" (Beat RATioS)
were designed, thanks to equations given by Gene Smith.
Bob Wendell found the unique WT with brats of either 1.5
or 2 on all major triads. But blind listening tests
showed the benefit of simple brats is subtle at best.

The latest work on the topic has revolved around using
JI tunings as WTs. Kalle Aho found the lowest subset of
the harmonic series which, when interpreted as a WT, has
no "harmonic waste" (Jorgensen's concept). I followed
with a [2 3 17 19] "constant structure" that functions
as a WT...

! 17/12-----17/16-----51/32
! .'/ \ / \ /|\
! 68/57/ \ / \ / | \
! | / \ / \ /57/32\
! |/ \ / \ /.' '.\
! 4/3-------1/1-------3/2-------9/8
! '. .' '. .' '. .'
! 32/19-----24/19-----36/19
!

I found this manually and suspect it is optimal, in terms
of providing the greatest number of 19-limit consonances
without having any 3rds sharper than 24/19 or any 5ths
outside of the 3249/2176 - 3/2 range, but I have it as an
open project to write software to verify this.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > Dear Carl,
> >
> > What I liked about this approach is that your can rate the
> > importance of intervals and keys and then the model computes
> > you a well-temperament. If there are similar or even superior
> > approaches to this, could you perhaps share some pointers to
> > literature or software.
> >
> > Thank you!
> >
> > Best
> > Torsten
>
> John deLaubenfels' software could minimize the squared
> error over an arbitrary list of intervals *for a given
> piece of music* (MIDI file). These are the so-called
> COFT (Calculated Optimum Fixed Tuning).

I wrote some time ago a little Mathematica code to numerically
minimise a weighted measure of deviation over 12-note tunings which
can easily be changed and extended to have arbitrary weights over
arbitrary intervals... with a little tweaking of the weights it spits
out certain 'historical tunings' rather easily. It's not rocket
science. I'm not sure how it helps in the end because you can get
almost any answer you like by manipulating the weighting.

Perhaps it could be useful if for example you wanted a tuning with a
certain known distribution of thirds and wanted to find the best way
to arrange the fifths to achieve that.

Now is this 'optimal'? -
http://eamusic.dartmouth.edu/~larry/mp3_files/owt_bach_wtc_examples/Fm_prelude_bk_1/harpsichord/septWT1.mp3

It's certainly extreme. I think their 'OWT2' is the only thing I'd
recognize as something like a musical tuning (more specifically, it
strongly resembles Kirnberger 'II' transposed a fifth sharp so that
the bad fifths are at A-E and E-B). How exactly they came to it is
rather unclear from the paper.

Their E major fugue is plagued with quite a few wrong notes. And
funnily enough they don't include any useful pieces in flat major keys
in their tests. The F major from part 2 hardly counts because it has
no sustained harmony.
~~~T~~~

Tom Dent wrote:

> http://eamusic.dartmouth.edu/~larry/mp3_files/owt_bach_wtc_examples/Fm_prelude_bk_1/harpsichord/septWT1.mp3

Sounds to me like meantone from F to A#, certainly not anything circular.

Petr