I previously dumped some thoughts

Found at: sdf.org:70/users/oms/interval--details.txt

I previously dumped some thoughts about learning relative pitch (and
less successfully, absolute pitch).

Here are some more beginner's-mind thoughts:

The specific interval methods I've seen all so far present specific note
pairs played on a piano. Identifying what I just heard presents a
two-part challenge that my mind has difficulty disentangling: the
absolute pitch challenge of identifying the specific note and the
relative pitch challenge of indentifying the interval. The pair "F4, C4"
sounds different than "C4, G3", even though they're both fourths down.

Is there a way to remove the absolute pitch component and generate a
sound pair that represents a platonic concept of "perfect fourth down"?
My first stab at this was to generate a base noise of 60Hz and every
integer overtone above [1], followed by the same type of noise at a
different base frequency, e.g. 44.95Hz for the (equal-temperament)
fourth down. The choice of 60Hz:

- is likely recognizable to North American audio engineers,

- isn't itself an in-tune note, sitting almost exactly halfway between
  B and B-flat if A=440,
- is near a sweet spot where:

  - if the base frequency gets too much lower, the overtones become so
    noisy and the time it takes for the signal to repeat so long, that
    my ear hears it not like a signal that has interesting
    frequency-domain content, but is instead something that is only
    interesting in the time domain. Almost like it's no longer
    "tone"-y, but more "drum beat"-y.

  - if the base frequency gets too much higher, it loses the property
    that the overtones create such a mess of tones in the space of
    pitches that it's not fully recognizable as one particular note.
    Consider that if we pitch it down slightly to 58.27Hz, the actual
    A#0, the first few overtones land at:

    A#0, A#1, F2, A#2, D3, F3, G-G#3, A3, C4, D4, D#-E4, F4,
    F#-G4, G-G#4, A4, A#4, B4, C5, C#5, ...

    (Update: the interval still sounds recognizable and has even less
     of a pitch by itself if we strip out the fundamental and the
     first few (3) overtones, and start on the D3.)

    Note that in the 4th (and unlisted 5th) octaves there is such a
    large mess of notes that it's hard to pin the mass to a particular
    root; in this case, A#. If we went up by an octave, then some of
    this mess will go out of the range of hearing, or out of the range
    of frequencies that we subconsciously consider when determining
    what the root of a note is from its overtones.

Fun fact: Vorbis does a profoundly bad job compressing these.

The result of using these to train myself on intervals is yet
undetermined, and likely couldn't become part of a self-experiment to
compare learning speed or interval classification ability to intervals
that have well-defined notes.

I also generated some sounds that are invariant under an octave shift,
so that C3 sounds identical to C4. These are philosophically similar to
the above in that I can use them to try to train myself to recognize
something as being "C" and not "C3" or "C4", with the idea that maybe
this would be easier to identify than piano notes. Coming soon: how I
did this.

[1]: Each with a random phase. Time-frequency experts will note that
     an impulse train at 60Hz has this property. This impulse train is
     a demonstration of where the casual assumption that the ear is
     phase-agnostic breaks down. The same experts may also notice this
     will still be a signal that repeats itself 60 times a second.
     It's almost the same as 1/60 of a second of white noise repeated,
     except that the white noise will also have harmonics at randomly
     higher or lower amplitudes.