What is the smallest countable limit ordinal in which ‘lost melodies’ occur

The question is in the title. This question is in response to the following paragraph found at the end of Prof. Hamkins’ answer to my MathOverflow question, Are ITTM’s necessary to compute Turing’s "computable numbers" and what dos that mean for ordinary recursion theory:

Regarding infinite time Turing machines, these are concerned frankly with much larger ordinals than $omega$, and their consequences and effects are realized on a much larger scale.