Electronic – Gain of Non-Inverting Amplifier Does Not vary with Resistor Values

Your problem may be that the LM358 is not designed to drive a speaker as a load. You will need to add an output stage with a low output impedance of ~4 to 32 ohms or less, depending on your speaker's impedance. Here is an example of a typical configuration using BJTs as the output stage:

Note how the feedback loop has been modified to incorporate the output stage. A trimpot is used in this schematic to adjust the volume.

My second hypothesis, is in the event that you have a high impedance set of headphones (around 600 ohms), is that your gain is too high and your are railing out. If that's the case, then you can either reduce your gain or replace your gain resistors with a trimpot and try adjusting the volume until it sounds better. If the "rett rett" sound you described is a loud sound, then I would recommend skipping my first suggestion and try this first instead.

You can't get more voltage out of an op-amp than it's supply voltage.

^{simulate this circuit – Schematic created using CircuitLab}

You may consider the output stage of the opamp as a pair of transistors with their emitters connected to the output.

If Q1 turns on it pulls Vout towards V+. If, instead, Q2 turns on the output will be pulled towards V-. It should be clear from this that the maximum possible output voltage is V+ and minimum is V-. In practice most op-amps can't even manage this but some can and these are called "rail-to-rail" type but the pull-up and down are weak and can't drive much of a load.