In a comment Thomas Blair said: "The way CPI is calculated has changed, which makes comparisons between eras or even years irrelevant."
An anonymous commenter writes "Your reasoning is so specious. Firstly you have to recognize that inflation (CPI) is measured differently from the 70s so you're comparing apples to oranges."
The BLS provides CPI data here going back to 1978 using the new CPI-U-RS, which allows for a comparison of inflation calculated in the late 1970s using the old CPI (CPI-U) to inflation calculated using the new CPI (CPI-U-RS), see top chart above for a comparison of the two different CPIs between 1977 and 1998 and bottom chart for a comparison of inflation rates from December 1978 to December 2007 (blue line is inflation using the old CPI and red line is inflation using the new CPI).
Yes, it is true that the old CPI overstated inflation (see top chart above) and it is true that inflation was overstated in some years using the old CPI (see 1979, 1980 and 1981 in bottom chart, when the blue line is above the red line), but then actually understated inflation in the 1983-1984 period (blue line is below red line), and was about the same using either method for most of 1982 and in 1985-1986.
Bottom Line: The old CPI slightly overstated inflation, and revisions were made to correct for the slight upward bias and make the new CPI a more accurate measure of inflation. Some reports seem to suggest that the new CPI creates a downward bias for the way inflation is now reported, and that it would be higher today if it was calculated using the old, upwardly biased method. I think it's better to think of today's CPI and inflation rates as being better, more precise and more accurate measures than those in the past, and any adjustments should be made to previous years' inflation, not today's rates.
Further, I don't think the differences in the old CPI-U and the new CPI-U-RS are meaningful in any way that contributes to the discussion about comparing today's inflation to the 1970s inflation. It's far from a specious, irrelevant, apple to orange comparison, it's much more like comparing one variety of apples to a newer, better variety of apple. Inflation in the late 1970s was at double-digit level by either CPI, and inflation today, measured more accurately than ever before, is nowwhere near double-digits.