The transformer seems a very simple object, basically it is made by two coil of wire winded around a steel core. The two windings are called primary and secondary.
Its task is to transform the amplitude of the signal to the ends of its windings without modifying the signal itself. But, something happen during this process, and we can achieve some cool Tone from this.
An ideal transformer will just transform the signal without modify it, but a real transfomer has some limits due the loss in the windinds, the resisntance of the copper, the capacity and the inductance of the winding himself, plus the interaction between the windings.
A complete description of the behavior of an audio transformer would be too long and complex to deal with here. The important concept to understand is that the geometry of the winding greatly affects the frequency response of the transformer itself and therefore the sound we get.
We worked on this for design out transformers and let them sound as we like. Look at the frequencies response of our FJOD, bold line is with the output transformer, thin without it. The two lines are not parallels, you can see a presence boost on the mids [400Hz-2kHz) and a cut on the hight treble; plus a difference output level due the ration between the primary and secondary winding.
A different frequencies response mean a different harmonics contents, let's understand what is distortion and how we can measure and undestand how the transformer acts on it.
Harmonic distortion is the process that adds overtones (harmonics) that are whole number multiples of a foundamental frequency. This is what happen inside a tube amp or inside a pedal in many way. We can generate distortion pushing the amplitude on some point of the circuit or with some devices like diodes or leds for accentuate the clipping. We can visualize the distortion with the FFT (Fast Fourier Transform) that let use see the harmonic contents of the circuit referred to one frequency called foundamental (usually 1KHz). The armonics are multiply of the foundamental. Even are 2°, 4°, etc (so 2KHz, 4kHz, etc); Odd are 3°, 5°, etc.
FFT @1kHz response of our FJOD, with the Output Transformer we have a bit more of the even harmonic (2nd, 4th. 6th) than without. This mean richer and more complex sound.