Consider the definition of "lost melody" given by Merlin Carl in his arXiv preprint, "[The Lost Melody Phenomenon](https://arxiv.org/abs/1407.3624v5)" (arXiv: 1407.3624v5 [math.LO] 16 Mar 2015):



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> A lost melody is a real number $x \subseteq \omega$ which is recognizable, i.e., for some ITTM program $P$, the the computation of $P$ with $y$ on the input tape (which plays the role of a real oracle for an ITTM) is defined for all $y$ and outputs 1 iff $y = x$ and otherwise outputs 0, but not computable, i.e., no program computes the characteristic function of $x$.
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Furthermore, Prof. Carl understands the notion of ITTM as follows:



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> These machines, which are basically classical Turing machines equipped with transfinite running time….
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Hamkins and Lewis define the conditions under which ITTMs can be considered "ordinary Turing machines" and in such manner show that classical computability theory can be deemed a subtheory of ITTM-computability theory:



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> Thus, restricting to the case of finite input and time, a function $f: 2^{\lt {\omega}} \rightarrow 2^{\lt {\omega}}$ is $\omega$-computable exactly when it is computable in the Turing machine sense.
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With this in mind I will rewrite Prof. Carl's definition without mentioning ITTM's:



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> A lost melody is a real number $x \subseteq \omega$ which is recognizable, i.e., from some Turing machine program $P$, the computation of $P$ with $y$ on the input tape (which plays the role of a real oracle for a Turing machine) is defined for all $y$ and outputs 1 iff $y = x$ and otherwise outputs 0 but not computable, i.e., no Turing machine program computes the characteristic function of $x$ [note that such oracles are standard in classical computability (recursion) theory—my comment (see Hartley Rogers Jr's text, chapter 9.2, "Turing Machines with Auxiliary Tapes", pg. 130)].
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All that remains is to note that both $\{x \mathrel| \text{$\varphi\_{x}$ is total}\}$ and its complement are both productive sets, and not recursively (computably) enumerable (which means, to me at least, not writable by any recursive function).


So can the singleton $\{\{x \mathrel| \text{$\varphi\_{x}$ is total}\}\}$ be decidable by Prof. Carl's oracle (restricted to classical recursion theory)?


I believe it can for the following reason: i) if one assumes that the oracle in question has multiple oracle tapes—$2^{\aleph\_0}$ of them in fact (one tape for each set of natural numbers)—when one compares the elements of $x$ with the elements of $y$ by the oracle, it will, in fact, show that $y \neq x$ in a finite number of steps, but only show that $y = x$ if the oracle machine does not halt. Now if one wishes to consider the oracle machine an ITTM (for the reason that if $y = x$ the the oracle machine does not halt), by the limsup rule the oracle machine must output a 1, but this is a trivial use of an ITTM.



Can $\{x \mathrel| \text{$\varphi_{x}$ total}\}$ be deemed a "lost melody" relative to classical recursion theory?
If we disregard the model of computation we want to use, a *lost melody* is a decidable singleton $\{x\}$ such that the point $x$ is non-computable. Classical computability theory cannot admit any lost melodies in $2^\omega$ because there are no decidable singletons in the first place, and it cannot admit lost melodies in $\mathbb{N}$ because there are no non-computable points.


We could weaken the requirement (as you may be doing in your question), and only ask for $\{x\}$ to be $\Pi^0\_1$, meaning that we can recognize the complement of $\{x\}$ but not necessarily $\{x\}$ itself. With $2^\omega$ as ambient space, this still doesn't give us any examples. However, if use $\omega^\omega$, then we see that every hyperarithmetical Turing degree has a representative $x$ such that $\{x\}$ is $\Pi^0\_1$.


Another way to get lost melodies without using a more powerful model of computation would be to move towards computable analysis, and to use general *represented spaces* as ambient spaces. This allows us trivial constructions such as considering any singleton $\{p\}$ for $p \in 2^\omega$ as ambient space in its own right, and since $\{p\}$ is trivially a decidable subset of $\{p\}$, we have plenty of lost melodies here. On the other hand, "nice" represented spaces would generally be computably separable, which immediately implies that every computably open (ie $\Sigma^0\_1$) singleton consists of a computable point.


