Fix crackle in sonify.time_frequency

[rabitt]

This addresses @craffel's comments on #224.

The main changes are:

1. Replaces @stefan-balke's asymmetric ramp between frames with an interpolator so the amplitudes will change smoothly (and symmetrically) over time.
2. Fixes a second cause of noisiness/buzzing I found in _fast_synthesize. The line:
   n_samples = int(10.0 * fs / frequency)
   is meant to generate 10 periods of the wave at f0=frequency, and the generated wave is copied. For non-integer values of frequency, 10. * fs / frequency is not necessarily an integer, so casting it to an integer results in generating less than 10 periods of the wave, and then there is a discontinuity when the wave is copied. I did a hack to fix this problem and force n_samples to always be an integer.

If this is merged, #224 can be closed. cc @stefan-balke

[rabitt]

@craffel I think this is good to go (on my end).

[bmcfee]

Doesn't this mean that only integer frequencies are sonifiable?

[rabitt]

No -- but i'm rounding each frequency to the nearest 10s place, so it might have an audible effect on higher frequencies.

[bmcfee]

offline conversation summary: might be better to make the number of decimals a parameter n_dec, and maintain a table of size 10**n_dec * fs so that a user can control the precision of the approximation.

[craffel]

Thanks for picking this up! Two small comments.

[craffel]

Looks like weights is unused? Did you mean to replace gram_interpolator(np.arange(start, end)) with weights below?

[craffel]

Can you replace all this logic with

gram_interpolator = interp1d(
    time_centers, gram[n, :],
    kind='linear' if len(time_centers) > 1 else 'nearest',
    bounds_error=False, fill_value=0.0)

or maybe 'zero' instead of 'nearest'? Alternatively if you can provide a little more of a comment for special casing, would be helpful for understanding it.

[rabitt]

@craffel I ran into a snag in the interpolation when len(time_centers)=1.

Using kind='nearest':

gram_interpolator = interp1d(
    time_centers, gram[n, :],
    kind='linear' if len(time_centers) > 1 else 'nearest',
    bounds_error=False, fill_value=0.0)

it fails because when kind='nearest' it expects time_centers and gram[n, :] to have more than 1 point.

Using kind='zero':

gram_interpolator = interp1d(
    time_centers, gram[n, :],
    kind='linear' if len(time_centers) > 1 else 'zero',
    bounds_error=False, fill_value=0.0)

The interpolator always returns zero -- for example:

f = interp1d([1], [3], kind='zero', bounds_error=False, fill_value=0.0)
f(-1)
>> array(0.0)
f(0)
>> array(0.0)
f(1)
>> array(2.1904428788e-314)
f(2)
>> array(0.0)

What do you want the expected behavior to be? For a length n output, do you want the same set of pitches to be sustained, or something else?

[craffel]

Hmm, those are frustrating behaviors of interp1d. Oh well, you can go back to what you had!

[craffel]

Thanks @rabitt !