This might explain strictly the math of beats in music:

"In **takt, integer values refer to beats, hence the value `2' refers to the second beat. Fractional values refer to moments within beats. For example, the value `3.5' refers to the second half of the third beat. In 4/4 meter, the value `4.75' refers to the last sixteenth of the measure. Up to two digits are permitted following the decimal point."

http://www.google.com/search?q=cache:s6Jftl8tFac:www.music-cog.ohio-state.edu/Humdrum/representations/takt.rep.html+"beat"+takt&hl=en

There really isn't a zero in music. The conductor prepares the orchestra before the first sounding instruments play that first beat, but the sound begins on the ictus, or downbeat. That first ictus of the conductor's baton is the visualization of the first beat the conductor intends to be sounded at that point, or very slightly thereafter in that conductors conduct ever-so-slightly ahead of the birth of the beat.

Long ago I remember reading words for the upbeat--and other words related to the beat of the baton, but I failed to learn them. If anyone knows these rare words, I would be very interested in reading them here.

But I thought that takt might be called for here, just as a side point, about how the math of beats appears.

Beat regards,
WoodWind