I simply point out that it is conceivable that both of these statements are correct. Tifft originally found a series of locations for a frame in which all sky periodicity is seen. He choose one near the galaxy frame (bearing in mind that at the time this was not known that accurately). Later he chose one near the CMBR. From my cymatics studies I can confirm that although such things are a bit counter intuitive, they can be true.
Not sure that I understand this. Both have studied redshifts in galxy pairs and small groups. AFAIK Arp has not looked at whole sky systems or frames that achieve this.
When a tight group is studied then there is no need to look at the frame used as all the objects are similarly affected by our motion. Many more surveys have been done with such samples and report 72 km/s periodicity. Only when whole sky studies have been done is a frame needed. Tifft has done much work on this and I already mention in the list part A which other studies repeated this type of work.
I have done this already in section A. Most studies simply say "galaxies". Tifft does mention "dwarf galaxies" as noted, and this study does show smaller harmonically realted periodicity, consistent with HT expectations.
Most of the samples are not large, but this simply means that greater consistency is required to achieve significance. A level of significance of .001 is equally valid in a sample of 200 as in a sample of 20,000. But of course bigger samples are better because there is the possibility of getting .000001 significance.

In most cases no stated errors are given for the periodicities. Tifft states that 72.135 km/s has an accuracy of close to .01 km/s but later changed the value to 72.45 km/s which may have made him wary of quoting error margins after that. Croasdale does examine this issue thoroughly and concludes that the 72.45 km/s value of Tifft has an accuracy of about 0.15 km/s. That places it 1 s.d from HT prediction of 72.153 km/s. I think that the Croasdale is a model one for statistical treatment (and this is an area that I have some expertise). He uses Monte Carlo methods to establish just how likely apparent periodicities are to appear in samples. He considers carefully the use of degrees of freedom and whenever he can uses parameters derived from past studies (such as Tifft's frame) in testing new frames. He further considers that with better data the frame may need to move a little.
He quotes separate significance for the 36 and 72 km/s periods. However he also looked at the interrelationship between the periods. For example, when a 12 km/s period can be found with a 72 km/s one then it can help to get a tight fix on the 72 km/s one.
Yes, there are variations in the findings, although the 72 km/s is nearly always found. The individual differences may be due to several reasons, and I can only mention on some of them. Tifft has shown, and I agree with this, that the accuracy of redshifts is very important because an uncertainty of 25% of the period will cause the peaks to be washed out. That means that with an accuracy of say 10 km/s you can detect the 72 km/s period but not the 36 km/s and smaller ones. One of the papers (Arp?) has a look at this issue by comparing several surveys measurements of the redshifts of the same galaxies and finds that accuracy is getting to around 3 to 9 km/s. This means that the 12 km/s period is very often going to be not found even if it is real.
Are these smaller galaxies? If so, then I would expect the periodicity to be a smaller cycle (like 12 km/s or 6 km/s) and the data may not be accurate enough to detect this. Tifft is clear about the need to work with different classes of objects separately. I would concur with this from a theoretical point of view in HT. If the data had unlimited accuracy this would not be a problem. Smaller irregular galaxies in the local group do seem to show a spacing of ~180,000 LY or 50-60 kpc (or multiples) from larger galaxies and each other. I have not seen an analysis of this published, but have observed this of the data. This observation is partly derived from a variety of sources. There will not be time to find all the references but want to raise this as something to watch for because such a common spacing is useful in the cosmic distance ladder.
By the consistency of the periods found. The 72 km/s period is found in a faviety of ways and is pretty solidly supported by many studies. The 36, 24, 18, 12 and 144 km/s periods have been found varying numbers of times each but look to be real enough even if not able to be found every time.
I have not seen any paper that uses Tifft's method (of a rest frame) and does not find at least one of the periodicities. I cannot remember any study of small groups of galaxies (or pairs of similar) that uses redshift only (not correlation functions in 3D) that does not find at least one of the periods mentioned, and nearly always the 72 km/s period. Certainly if there are such papers they are few and far between compared to those that do find the periods/
I totally agree that the newer data is a great resource for this. One of the questions that I asked was how to download some of this data without getting gigabytes. I did succeed in getting some, and doing some analysis of smallish samples. Looking at the redshift differences between galaxies in one part of the sky, it does show peaks at 72, 144 and 216 km/s (and some other places). Having established this, I think that I can go on to get further samples from various directions over the sky and look really hard at the issue of frames in which the various periodicities are universally present. I also want to look at different categories of galaxies to do this, such as dwarf galaxies. Obviously this needs redshifts accurate to around 1 km/s to get the best results, and a smaller sample is more acceptable than less accurate data. I need to learn more about galaxy classification so that I select samples sensibly. However I expect to be able to get results that are not p<.05 or p<.01 but more like p<.000001 which I think are more likely to make people have a hard look. If periodicity is real then with more data such significance must be achievable.
Unfortunately I cannot access this without paying. However I can read the abstract and agree with it. As a lone researcher this is a potential problem and for a group it is still true. That is why p<.000001 results are so useful because if you get a string of them they show that you are either on to something or doing millions of studies.
As mentioned, most papers just state "galaxies". However there may be selection of types happening for a variety of reasons. I would think that good redshift measures are easier for brighter objects. Therefore larger / brighter galaxies are more likely to dominate at the limits of surveys while dimmer / smaller ones may dominate at the closer range of a sample.

I agree that this is an important point, and Tifft certainly seems to be on to it in his later papers even if everyone else is not.

I just note that a couple of redshift periods are produced in just one paper and then not found again. These probably are really just chance statistics as in a hundred papers we would expect 1 to 5 false positives at the .01 to .05 level. This sort of thing is certainly true in cycles research, and that is why the method that Dewy uses of collecting many periods and looking for common ones is a good idea. When they also have a common phase in seemingly unrelated things then you know that it is a real phenomenon. I would really like to see more scientists take a wider interest in cycles because the interdisciplinary connections are so great. From the 1930s there were a series of interdisciplinary cycles conferences which continued when Dewey and others formed the FSC in 1941, but following Dewey's death in the late 1970s this sort of thing died out. But even cycles study has cycles in it and it seems there might be a resurgence coming.

Geology has recognized the relevance of long astronomical cycles such as the 410,000 year orbital change cycle (Milankovitch cycle) in affecting climate. If astronomy also recognized the geological cycles as related to the large scale structure periodicities then it would work well the other way also in getting an accurate fix on large scale periods. This in turn would allow very precise measures of the rate of evolution of the universe over cosmological time.