Decreased left ventricular ejection fraction is the most commonly used risk factor for identification of patients at high-risk for lethal ventricular arrhythmic events. Twenty-four-hour electrocardiographic (ECG) approaches to risk stratification include: counts of ventricular premature contractions (VPCs), measures of heart rate variability (HRV), and heart rate turbulence (HRT) which has two components, turbulence onset and turbulence slope (TS). Refinement of these ECG risk stratifiers could enhance their clinical utility. We explored the structural relationships between heart rate (HR) and HRV and HRT measures. Our goal was to separate out the component of these measures due to the underlying average heart rate (HR), thus potentially reducing the variability of the measures and increasing their power to stratify risk. We proposed re-scaling tachograms of heart-beat intervals so that the re-scaled tachogram has a HR of 75 (or equivalently an average interval of 800 ms) and calculating HRV and HRT from the rescaled time series. We also explored the relationship between the number of VPCs and HRT. We showed that TS is structurally related to the number of VPCs (and hence to the length of the ECG recording). We proposed an adjusted TS that is independent of the number of VPCs. We also addressed the ability of shorter ECG recording to estimate HRV and HRT measures. We evaluated standard and rescaled HRV and HRT measures using qualifying ambulatory ECG recordings from 744 patients in the Cardiac Arrhythmia Suppression Trial. We found that measures based on the rescaled tachogram had reduced variance (20% to 40%). Correlations between measures were also substantially reduced. We also found substantial circadian effects on some, but not all HRV indices, not explained by the circadian pattern in HR and possibly pointing to additional measures for risk prediction. In conclusion, we found that adjusting for HR and the number of VPCs in heart-beat related ambulatory ECG measures has the potential to significantly improve the power of these measures to risk stratify cardiac patients.