The entropy S in natural time as well as the entropy in natural time under time reversal S_have already found useful applications in the physics of complex systems, e.g., in the analysis of electrocardiograms (ECGs). Here, we focus on the complexity measures Λt which result upon considering how the statistics of the time series ΔS [≡ S - S_] changes upon varying the scale l. These scale-specific measures are ratios of the standard deviations σ(ΔSt ) and hence independent of the mean value and the standard deviation of the data. They focus on the different dynamics that appear on different scales. For this reason, they can be considered complementary to other standard measures of heart rate variability in ECG, like SDNN, as well as other complexity measures already defined in natural time. An application to the analysis of ECG - when solely using NN intervals - is presented: We show how Λt can be used to separate ECG of healthy individuals from those suffering from congestive heart failure and sudden cardiac death.