By the 1960's the instability strip had been observed to have the following general properties: periods for Cepheids ranged from several days to a few hundred days; at constant period, the B magnitude total width of the PL relation was about 1.2 mag; the V magnitude width was measured to be about 0.9 mag; and the (B-V) color width was found to be about 0.4 mag, with the reddest Cepheids being the faintest at any given period. In a practical sense, this meant that in estimating distances, any individual Cepheid could deviate from the statistical ridge line by up to ± 0.6 mag in B; and such an error (if applied to one Cepheid) would translate into an equivalent error of about 30% in distance. Large samples can decrease the error on the apparent modulus inversely with the square root of the number; a formal error of only 10% being possible with a sample containing as few as a dozen Cepheids.
The discussion in the preceding section concerned an idealized PLC relation, expressed in its linearized form. Some of the difficulties encountered in the empirical calibration of this relation will be discussed in the present section. We concentrate on extragalactic studies. And so we do not discuss the issue of how best to determine independent distances and independent reddenings for the Galactic population of classical Cepheids: as plentiful as they are, Galactic Cepheids in the field are problematic, while only a handful of (short-period) Cepheids are contained in open clusters, which are generally sparsely populated, and often heavily obscured. The interested reader is however referred to the recent papers by Fernie (1990), Fernie & McGonegal (1983), Feast & Walker (1987) and Jacoby et al. (1992) for a pathway into the literature on this Galactic approach to the calibration.
Because they are nearby, and because of the large numbers of Cepheids cataloged in them (Payne-Gaposchkin 1971, Payne-Gaposchkin & Gaposchkin 1966), the Magellanic Clouds have long been the testbed for calibrations of the period-luminosity and period-luminosity-color relations. Indeed, the original discovery of these relations was made in the Magellanic Clouds (Leavitt 1906). These same large samples allowed the first estimates of the slope of the PL relation and first approximations to the period and color dependences of the PLC. Now with a number of analyses of the geometric expansion parallax to the LMC via the remnant of Supernova 1987A there are geometric distance modulus estimates of 18.50 0.13 mag (Panagia et al. (1991), 18.38 ± 0.07 mag (Schmidt-Kaler 1992), 18.52 ± 0.13 mag (McCall 1993), 18.61 ± 0.11 mag (Crotts et al. 1995), < 18.37 ± 0.04 mag (Gould & Uza 1997) for the LMC. While the precision of this measurement might be expected to improve with time as the expansion continues to be monitored, the systematics (assumptions about ring geometry and placement of the SN with respect to the main body of the LMC, etc) will soon dominate the solution. Nevertheless, it is reassuring that the majority of the determinations agree extremely well with alternative estimates of the LMC distance modulus, and that an independent check of the zero point (at the 10-15% level) is provided by the measurement of RR Lyrae distances (e.g., Reid & Freedman 1994), which also are in good agreement with the Cepheids (for more recent reviews see Westerlund 1990, 1997 and references therein).
For work on the distance scale, the existence of a statistical relation between period and luminosity is of such great utility by itself, that it is of little wonder that concern about second-order effects in the calibration (i.e., those aspects above and beyond establishing the slope and zero point) were not of immediate import in the earliest studies. Some of these issues are: the origin of the scatter in the PL relation; the systematic effects of reddening; the systematic effects of metallicity (e.g., Gascoigne 1974; Stothers 1983; Freedman & Madore 1990), companions (e.g., Madore 1977; Coulson et al. 1986); CNO abundance, helium abundance and mass loss (e.g., Lauterborn et al. 1971; Lauterborn & Siquig 1974; Cox et al. 1978; Becker & Cox 1982); magnetic fields (e.g., Stothers 1982); the possibility of curvature in the PL relation (e.g., Fernie 1967, Sandage & Tammann 1968, 1969); the relative disposition and the slopes of the red and blue edges of the instability strip (Fernie 1990), and the physical origin of these constraints (e.g., Iben & Tuggle 1972a, b; 1975; Chiosi, Wood, Bertelli & Bressan 1991), etc. Unfortunately, several of the corrections to be considered are probably manifesting themselves simultaneously, and at the same level of numerical significance.
Before we can approach an empirical determination of the coefficients in the PLC (or any determination of their variation with metallicity) we must solve the reddening problem. While theory predicts a finite width to the instability strip (with temperature/color being the controlling parameter), and while metallicity is a quantity that is known to be different from galaxy to galaxy (and it is known to vary systematically within individual galaxies), only when reddening has been accounted for can we go on to look for meaningful correlations of luminosity residuals with intrinsic color and/or metallicity, for instance. To decouple and solve for the effects of metallicity, reddening, and intrinsic temperature variations, high precision photometry is a prerequisite, and at least as many independently measured quantities are needed as there are parameters to be determined.