I think she's got it slightly wrong. It's not that one of the 3/4 bars gets an extra beat - it's the previous bar that loses a beat.

https://www.youtube.com/watch?v=AVo0Q8G8tS8&feature=youtu.be...

Listen to the bit after she says '4' - it sounds syncopated in her counting. (Actually - 'syncopated' might not be the correct word - what I mean is that the '1' ceases to sound like the start of the phrase)

It's a fermata on the second beat of that measure. Holding a note doesn't mean you're changing the time signature.

https://en.wikipedia.org/wiki/Fermata

It's definitely a fermata.

Arguing otherwise is like saying the national anthem changes to 15/4 when singers hold "land of the freeeeeeeeeeeeee."

Vi Hart makes a good case (in my opinion) that there is a change in time signature. It's clearly an extra beat, not a "hold of indeterminate length".

Except people do hold it for different lengths of time.

Yes, there may be many people who hold it for about the same length of time, but an extra beat's worth of time is by no means universal.

It's a fermata.

It may have a fermata in addition to the extra beat. The quarter-note (3/4) timing seems very rushed compared to how the song is currently sung in modern culture.

I'm quite sure the other measures are in 3/4 time. Here's[1] a fairly typical notation (first from google image search). The previous measure would be the descending D-B-G quarter notes with the lyrics "birthday dear" (two beats for "birthday", one for "dear").

The next bar, which is notated in that image as a F-E legato slur only notates the E as a quarter note. If that was played as it is notated, the part where you say the person's name would sound very rushed. We probably started holding the E to allow time for longer names, and it just stuck.

(I find it interesting that this image doesn't even include a fermata[2] on the E; it includes the triplets on "hap-py" which is not in the original, but rushes the name.)

Also, there is a slight syncopation-like effect when she says "4" - that's the triplet effect where three notes are played in the time of two (in this case a dotted-eighth + sixteenth taking (2/3 + 1/3) of a quarter note in time). That's after the extra beat.

[1] http://abcnotation.com/getResource/downloads/image/happy-bir...

[2] https://en.wikipedia.org/wiki/Fermata