Optimising synthesizer tuning

After Bruce Gilson showed interest in my mode-fitting algorithm it
suddenly occurred to me that the same method can also be used for
tuning a synthesizer. To do so, the closest mode is calculated for the
octave division equal to the tuning resolution of the synthesizer. The
case is that a better quantisation can be made than just taking the
nearest step for each pitch. Just like the methods record companies
invent to squeeze everything out of the 16 bits of audio on a CD, like
Super Bit Mapping and so on, this method gives a better mapping for a
tuning system.
Take the following contrived example. For simplicity we assume a
tuning resolution of 1200 steps per octave.

0: 0.00 cents
1: 100.55 cents 100.55
2: 200.55 cents 100.00
3: 300.55 cents 100.00
4: 700.45 cents 399.90
5: 800.45 cents 100.00
6: 900.45 cents 100.00
7: 1200.00 cents 299.55

Doing ordinary roundoff this gives the following number of steps:

1: 101 101
2: 201 100
3: 301 100
4: 700 399
5: 800 100
6: 900 100
7: 1200 300

The disadvantage we are seeing is that interval 3-4 is 399.9 cents but
it gets only 399 steps of 1 cent.
The solution is to give up the fixed tuning of pitch number 0 and
incorporate it into the roundoff process. If the grid of tuning steps
is shifted at most half a step up or down, pitch number 0 will always
be rounded off to zero but the other ones can be rounded off to
different values. The grid needs to be wobbled a bit in order to find
the optimal position. From a certain position, the next optimal shift
can be calculated and be fed back for a new round of rounding off.
This settles quickly. The variance is calculated with respect to the
shifted grid. For the example this gives as best (least squares) fit:

1: 100 100
2: 200 100
3: 300 100
4: 700 400
5: 800 100
6: 900 100
7: 1200 300

The grid is shifted is 0.4286 cents.
I'm currently trying to find an efficient way of best guessing a shift.
The method I had originally was found not to work in all cases.
For tunings with another octave than what the tuning steps of
synthesizers are based on (mostly 2/1), and not a multiple of it, this
method doesn't work.

Manuel Op de Coul coul@ezh.nl

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Regarding Partch transcriptions: Ted Mook, former Newband string player,
has recently released a recording of the 17 Li Po Lyrics and has re-notated
the score. I have his email address at home and will send it to you later.

A while back I did a MIDI version of Barstow but I guarantee a MIDI file makes
less sense than Harry's notation (plus the project was a total failure in
terms of musical sensibilities--)

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> It is well discussed that his tablature notation makes reading a score
> tedious

I hadn't thought of this before, but interpreting Partch's scores might in
itself be a reason to have copies of his instruments around the house. I
suspect that having them around might make it easier to relate to, for example
the chromelodeon parts.


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