# Re: [Fwd: technical notes on data used by Martin Braun]

```Paul Boersma wrote:

>Is it likely that the measured F0 values had a spacing of 1/4 semitone?
>It seems more likely that they were expressed as an entire number
>of samples per period.

I don't know about GIPOS, but it is true that many F0 estimation methods
measure 1/F0 with a resolution of one sample.  I made an experiment
inspired by Paul's praat script (translated to matlab).  Specifically I did
the following:

- Generate 500000 random frequency values uniformly distributed on a log
scale between A2 (110 Hz) and A4 (440 Hz), which is roughly the extent of
the data in Braun's paper.
- Convert frequency to period.
- Quantize based on a sampling rate of 16000 Hz.
- Convert to an octave scale relative to a reference value (440 Hz).
- Make a histogram with 1/4 semitone bin size.

As Paul found, the histogram has a very pronunced structure.  Some bins are
actually 0.  Folding the histogram into 4 bins representing counts modulo
one semitone also gives a clear structure.  If consecutive bins are paired
and averaged, the ratio between largest and smallest is about 1.11, not too
far from the ratio 480:416 (about 1.15) found by Braun.

If the statistics are limited to ACDEFG, the ratio is about 1.25, not too
far from the ratio 248:182 (about 1.36) found by Braun.

The exact values vary strongly with the reference value that defines the
boundaries of the 1/4-semitone-wide bins (which according to Bob Ladd were
not aligned exactly on 440 Hz).  They also depend on the overall shape of
the F0 distribution, and of course they fluctuate from trial to trial.

Interactions between successive quantization operations (a sort of moire'
effect) can produce a very salient pattern even if the underlying
distribution is smooth.  Of course this does not imply that such was the
case for the original study.  The surest way to verify would be to look at
the original data.

Alain

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Alain de Cheveigne'
CNRS/IRCAM, 1 place Stravinsky, 75004, Paris.
phone: +33 1 44784846, fax: 44781540, email: cheveign@ircam.fr
http://www.ircam.fr/equipes/pcm/cheveign
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```