Tifft's "periodicity" only shows up in small samples (<400 galaxies?). They are non-existent in the large surveys using more robust tests.
His periodicities are an artifact of performing power spectra analysis on approximately gaussian distributed data points. In this case, the error bar is equal to the amplitude, an issue understood by any engineer doing digitial signal processing when noise is present. This is simple to demonstrate: Create a line of gaussian distributed points around some mean value and do a power spectrum. You'll always get some peaks, they're just meaningless.
Because galaxies are roughly gaussian distributed around the line of the Hubble Law, v=Hd, this seems to be the trap Tifft fell into. Some have tried to claim that if we're offset from the center of the periodicity distribution, this will "wash out" the periodicity in large surveys.
The problem with that is if the periodicity were real, an offset from the distribution center will largely change the PHASE of the FFT. But power spectrum = FFT*(complex conjugate FFT) so phase information cancels out in power spectra.
Consider a periodic redshift around a center as a series of concentric spheres with some distance, s, between each shell. If you look along rays in random directions, your sample will have points separated by s along the ray. Now move slightly off the center of the spheres and look along the rays again. Nearby, the separation of the samples will be slightly different but the major difference between the rays will be the relative phase of the sampled points. As the rays sample the spheres surrounding the observer, these points will significantly outweigh the nearby points. The power spectrum will preserve the peaks, but they will be a little broader. I suspect you would have to be far from the center (perhaps on the order of half the radius of the universe) to wash out the peak completely.
This is a common problem with Tifft's analysis. I had to deal with a similar issue in my dissertation because I had gamma-ray data with a signal-to-noise ratio, S/N ~ 0.4. The signal was a fraction of the intensity of the noise in my data stream. I was also looking for periodicities. Any test for periodicity had to be tested with simulated data of similar S/N but with known properties/periodicities to see if the test could find them. Many proposed tests would fail these validations.