Does resonance still occur at half, a quarter, etc. of the resonant frequency? Examples of resonance that I have seen are pushing a swing and shattering a glass.
I know the swing analogy, that if you push at the right frequency, you can make the swing go higher and higher. My confusion stems from this reasoning that I have had:

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*When pushing a swing, instead of pushing it every time, if you pushed it every 2nd time it came back, it still works to increase the amplitude of the swing, i.e. creating resonance.

*So basically what this means is that by pushing the swing with half the resonant frequency still creates resonance.

*Then applying this to shattering a glass. If the glass had a resonant frequency of 880 Hz, it would mean that you need to sing the note A5 to shatter it. Using the conclusion from the swing example, this means that I should be able to sing the note an octave below, an A4 / 440 Hz note, and it would still work to be able to shatter the glass (although it probably would take longer as the glass is only having energy added to it every 2nd oscillation).

So while it's always like, an opera singer singing a really high note to shatter a glass, shouldn't it still work if you sang a note an octave below (because maybe you can't reach high enough for the real resonant frequency), or even two octaves (which would be equivalent to adding energy every 4th oscillation)?
However, since I've never heard of this, I get the feeling that I'm probably not understanding something and have some fallacies in my reasoning.
 A: A simple oscillator, such as a mass on a spring, oscillates, when displaced and released, with simple harmonic motion, that is the displacement varies sinusoidally with time at a natural frequency determined by the mass and the force constant.
It performs forced oscillations when subjected to a sinusoidally varying 'driving force'. The maximum amplitude (resonance) is when the driving force equals the natural frequency. [If there are light damping forces the resonance frequency will not quite equal the natural frequency.] There are no other resonance frequencies; the amplitude of the oscillations drops off at an increasing rate as the forcing frequency departs from the natural frequency.
So how about giving the swing a push at a sub-multiple of its natural frequency? Surely that will give a resonance-like effect? Yes, and it's easy to see why in mechanical terms. All the same, this is not resonance at this sub-multiple frequency, because (a) the forced oscillations are not at the frequency of the applied pulses of force, but at the swing's natural frequency, (b) resonance is defined for a sinusoidal driving force, and a sinusoidal force with this sub-multiple frequency won't give rise to any specially large response. [In mechanical terms, during some parts of the force's sine wave cycle the force will be pushing the swing 'the wrong way'!]
There is, though, another way of looking at what's going on ... The pulsed driving force can be Fourier-analysed into a sum of sinusoidally varying forces at the pulse frequency and multiples of it. One of these multiples or 'harmonics' will be at the swing's natural frequency, so we can, if we wish, regard the pulsed pushing as giving true resonance after all – at the swing's natural frequency.
The wine glass is not a simple oscillator but has a distributed mass with elements of the mass exerting forces on neighbouring elements and the elements able to move (slightly but enough) relative to each other. A stretched string provides a simpler example of such a system. Distributed mass systems like these can support standing waves, and have many natural frequencies and therefore many true resonance frequencies. For the stretched string the natural frequencies are multiples of a 'fundamental frequency'; for the wine glass the relationship between natural frequencies won't be so simple.
