Although deep neural networks can estimate the key of a musical piece, their supervision incurs a massive annotation effort. Against this shortcoming, we present STONE, the first self-supervised tonality estimator. The architecture behind STONE, named ChromaNet, is a convnet with octave equivalence which outputs a "key signature profile" (KSP) of 12 structured logits. First, we train ChromaNet to regress artificial pitch transpositions between any two unlabeled musical excerpts from the same audio track, as measured as cross-power spectral density (CPSD) within the circle of fifths (CoF). We observe that this self-supervised pretext task leads KSP to correlate with tonal key signature. Based on this observation, we extend STONE to output a structured KSP of 24 logits, and introduce supervision so as to disambiguate major versus minor keys sharing the same key signature. Applying different amounts of supervision yields semi-supervised and fully supervised tonality estimators: i.e., Semi-TONEs and Sup-TONEs. We evaluate these estimators on FMAK, a new dataset of 5489 real-world musical recordings with expert annotation of 24 major and minor keys. We find that Semi-TONE matches the classification accuracy of Sup-TONE with reduced supervision and outperforms it with equal supervision.