Asteroseismology probes the internal structures of stars by using their natural pulsation frequencies . It relies on identifying sequences of pulsation modes that can be compared with theoretical models, which has been done successfully for many classes of pulsators, including low-mass solar-type stars , red giants , high-mass stars and white dwarfs . However, a large group of pulsating stars of intermediate mass-the so-called δ Scuti stars-have rich pulsation spectra for which systematic mode identification has not hitherto been possible . This arises because only a seemingly random subset of possible modes are excited and because rapid rotation tends to spoil regular patterns . Here we report the detection of remarkably regular sequences of high-frequency pulsation modes in 60 intermediate-mass main-sequence stars, which enables definitive mode identification. The space motions of some of these stars indicate that they are members of known associations of young stars, as confirmed by modelling of their pulsation spectra.