I understand how some words have emphasis on certain syllables,, but from following the furigana in the following video:

I just can't understand how the singer comes up with the pronunciation of the kana during parts 0:59 - 1:02.

The kana reads: tsu yo ga ri wo hi to tsu kii te ku re
But it sounds like: tsu yo gar ri wo shko shkin de ku re

EDIT: added "yo" and "i"

Marble101 Wrote:The kana reads: tsu ga ri wo hi to tsu ki te ku re
But it sounds like: tsu gar ri wo shko shkin de ku re

You're missing a yo in both lines, and an i in "kiite":
tsu yo ga ri wo hi to tsu kii te ku re

In fluid speech, it's not uncommon for /hi/ to be pronounced more like /shh/ when the /i/ after it is devocalized. It's just what the mouth naturally does when you're talking fast.
And don't forget that /u/ and /i/ can be devocalized between unvoiced consonants and after an unvoiced consonant at the end of a word.

So I guess a representation of the line that's closer to how it's actually being pronounced here would be:

tsuyo gari wo shhto tskiite kure

The rest sounds perfectly like what's written there, and you seem to be just mishearing it. Try using a program like Audacity to slow the audio down and that should elucidate it for you.

What do you mean by "unvoiced"?
They just don't pronounce the vowel?

Yes. Any decent Japanese learning guide (including textbooks) should explain that. It's pretty important.

Also the "h" sound in "hi" is different from the "h" sound in ha, he, and ho. It can sound like "sh" although it's not the same sound. Pronouncing ひ as the English word "he" (i.e. he/him/his) gives you a foreign accent. (The IPA symbol for the consonant in ひ is [ç].)

And as JimmySeal says, devoiced vowels (or "whispered mora" or whatever term your book uses) is a very important concept. It's said to be a feature of Tokyo Japanese although I'm sure it's present in some other dialects as well. The basic rule is that a vowel is devoiced when it's between two voiceless consonants, or at the end of a phrase/sentence after a voiceless consonant.

I can hear exactly what is written there but then again my receptors are conditioned with a different sound system from yours so what I can hear doesn't have to be what you do.

It sounds exactly how it's written to me too... o_O

The 'h' in ひとつ is a voiceless palatal fricative. Pronounce it as in the English word 'hue'. Examples in other languages are on the Wikipedia article linked above.

Also useful: https://en.wikipedia.org/wiki/Japanese_phonology.