Planche, T. Baartman, R.A. JACoW Publishing Geneva, Switzerland 2673-5482 978-3-95450-212-7 10.18429/JACoW-CYCLOTRONS2022-FRAI02 English 356-358 cyclotron closed-orbit betatron proton resonance Contribution to a conference proceedings 2023 2023-10 https://doi.org/10.18429/JACoW-CYCLOTRONS2022-FRAI02 https://jacow.org/cyclotrons2022/papers/frai02.pdf We demonstrate that a cyclotron can be made to have precisely constant betatron tunes over wide energy ranges. In particular, we show that the horizontal tune can be made constant and does not have to follow the Lorentz factor gamma, while still perfectly satisfying the isochronous condition. To make this demonstration we developed a technique based on the calculation of the betatron tunes entirely from the geometry of realistic non-hard-edge closed orbits. The technique is applied to the particular case of a 800 MeV to 2 GeV proton cyclotron to produce a design that is presented here.