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There are lots of publicly available data sources that aren't searchable and should be.

For instance, why can't we search for melodies?

I would love to be able to sing a tune and have a system return a list of music pieces where that sequence of intervals is found.

It seems pretty possible to do with current technologies - index side: index n-grams of intervals from all music pieces produced (the total corpus of "documents" is probably less than 100 million, not a lot by today's standards) - search side: analyze frequencies, convert into n-grams, match those in the index

So why is it not done?

It would probably be hard for a random startup to do the indexing side (ie, gain access to the corpus of every song ever produced in digital form), but for Google or Amazon it would seem to be not that hard...?

well there are already a few apps such as musicaroid that searches Music and Video by using Yahoo Music(US) API, Google Youtube API, iTunes Store API. Than there are apps like Hound and Midomi where serach happens through voice & humming... and we can look forward to some great stuff when technologies like "Intelligent A.I. search' being developed by Ray Kurzweil at Google becomes a reality.
Hound is just "voice search" for songs: but I'd rather type than speak.

I had never heard of Midomi; just went and tested it. It seems to try to do what I'm talking about, but the "matches" seem completely random.

It may be that doing this is much harder than it seems; but what I'm talking about is really not about a specific performance, but about melodies (parts of melodies as sequences of intervals).

And it should work just as a text search engine works (matching a query to entries in an index); no special AI required for a first version.

I agree, it looks like a hard problem to me too... esp. given the fact that so many of the songs have same scale and chord progressions
'Shazaam' iPhone/Android app does nearly that... it can identify the song by listening to a part of the actual recorded song. But it still can't identify the one that you sing or play on an instrument... I am pretty sure they'll be working on something like this...
Shazaam doesn't do what I describe, it just tries to match part of a performance to the recording of that exact performance.

In my experience, it also doesn't really work.

Anecdotally, it works very well for me the last few years. On the other hand, Gracenote — which Path uses — has been terrible for me.

Does Shazaam not work well for you for a particular genre?

To be fair, I didn't test it extensively, but the couple of times I tried it (on French songs, mainly) it failed.

I also don't fully understand the use case, except maybe if you listen to a lot of radio in a car?

At home the music that comes out of my speakers is music I already own; sometimes I listen to radio but even then it's usually faster and more certain to just go to the station's website to check what music was playing a few minutes ago, than to be able to launch Shazaam before the song has finished playing.

Now if I could sing into an app that would be completely different.

Ah. I've only tried in on English (language) songs. It's handy if you are out in a cafe or otherwise, public area and want to identify a song.

Singing into the app would be cool. In my ex-company, there were folks working on getting it to work for Karaoke. So you can hum a song and it'll figure out the right track for you to sing with. I don't know if there's a commercialised product like this.

To shazaam's credit, it identifies popular songs even in non-english languages (Bollywood, Italian, several French songs; anecdotal evidence anyway)
yup, it only matches the exact parts but that it does that very well, It even identified a very old and unpopular song in Marvadi language (a minor dailect of hindi, spoken in some parts of Rajasthan, an Indian state). that must mean that they maintain a very extensive database...
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I don't want a "smart personal assistant" who is inherently corrupt, i.e. driven by the desire to sell ads and not to assist me in my best interest. Hopefully, the future of search will be more decentralized and driven by individuals' needs.
I think you fail to explain why one precludes the other. (WRGT ads vs a user's best interest). And I also think that you're overly focused on the monetization aspect of current search technology. Which neither search in general nor Google in particular are tightly coupled to. It's simply the best practical monetization method at the present time.
I do not think, showing me ads is a problem for me when I am using a free service. It is finally my choice to click them or not...