When the kids were little and we were at the kind of event where they might get separated from us, I'd write my phone number on tape and stick it to their backs:

Anna at NEFFA 2019, age 3

The kids are getting older now, and are playing independently enough that it would be good if they always had my phone number. Lily (9yo) has had a piece of paper in her backpack for a while now, but unless she's going to or from school she won't have it with her. Several rounds of "repeat after me" weren't working, so this weekend we made up a song for it. After just a few times singing it she has it down.

This may be a trick that works unusually well for Lily, since she's pretty musical, but the popularity of jingles for phone numbers in advertising makes me think this would work for others?

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My mom and her siblings report learning their phone number this way. It's effective enough that I know that the house phone of my grandparents half a century ago ends in seven.

Australia has a phone number you can call if you need help improving your literacy, and there were a bunch of TV ads advertising this number via a jingle that is catchy enough that I still remember it to this day (not having lived in the country since 2016). Plausible that that jingle was more optimized than most for memorability given their target market, and indeed I don't think I remember any other phone number jingles. If true, suggests that there continues to be returns to optimization for catchiness for quite a while.

I guess it could be that that hotline had a more consistent phone number and advertisement style than others.

Assigning digits to scale degrees feels like an interesting alternative here, especially for the musically-inclined.

You could do that, but I don't think it would sound very good? And I don't think it would make it easier for a kid to memorize?

I'm actually writing from experience there! That's how my memory still mainly encodes the few main home telephone numbers from when I was growing up (that is, I remember the imputed melodies before the digits in any other form), but it's possible they were unusually harmonious. I don't think it was suggested by family, just something I did spontaneously, and I am not at all sure how well this would carry more generally… it may also be relevant that I had a bunch of exposure to Chinese numeric musical notation.

How does that work with 10 available digits and only 7 scale notes? Do three digits become accidentals or something?

Maybe use more than one octave of range? So if we wanted to do it in Am we'd turn 123-456-7890 into A3 B3 C4 - D4 E4 F4 - G4 A4 B4 G3

I think with this system you will end up with too many large difficult and uncatchy jumps. Plus similar phone numbers will sound similar which is not what you want.

That's what I did, though with do on 1 and mi′ on 0 (treating it as 10). I'm not sure what similarity metric is most relevant here, but in case of near-collisions, small additions of supporting rhythm or harmony could help nudge some sequences apart; swing time would give added contrast to even vs odd positions, for instance. Anyway, it sounds like this might not generalize well across people…

Aside: this is a time when I'm really appreciating two-axis voting. The vote-signal of “that's mildly interesting but really doesn't seem like it'd work” is very useful to see in a compact form, even though the written responses contain similar information in more detail.

Works for UK emergency services (joke!): 

Another option: my father reports he usually memorizes phone numbers based on the geometric pattern they make on a typical keypad.