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Perl 5, Ruby, JavaScript (Chrome), PHP, Python 2, Python 3, 1206 bytes, score 6⁶/1206 = 38.6865671641791

q=1//2;s=+0;s|0;"/.__id__;#";_="""q=1//2;s=+0;s|0;"/.__id__;#";_={0}{1}{0};R=chr((39,34)[1/2>0])*12;Q=chr((39,34)[1/2==0])*3;q={0}{2}{0};print(_.format(Q,_,q[0:-12]+R,"%"))""";R=chr((39,34)[1/2>0])*12;Q=chr((39,34)[1/2==0])*3;q="""|;#<?ob_end_clean();
eval($_='"$t"?0?[$q=37 .chr+113 .chr+38 .chr,$p=(96 .chr+39 .chr)*4,$n=10 .chr,$s=39 .chr*12,$S=34 .chr*3]:eval("printf=console.log;c=String.fromCharCode;$q=c(96);$p=c(37,113,38,39).repeat(4);$n=c(10);$s=c(39).repeat(12);$S=c(34,34,34)"):[]?[$q=q(!g)^PA,$p=HARHARHARHAR^od2od2od2od2,$n=R^X,$s=bbbbbbbbbbbb^EEEEEEEEEEEE,$S=hhh^JJJ]:[$q=O^h,$p=str_repeat(RHa3^w9GS,4),$n=R^X,$s=str_repeat(b^E,12),$S=HHH^jjj];//#');printf($x='q=1//2;s=+0;s|0;"/.__id__;#";_=%7$sq=1//2;s=+0;s|0;"/.__id__;#";_={0}{1}{0};R=chr((39,34)[1/2>0])*12;Q=chr((39,34)[1/2==0])*3;q={0}{2}{0};print(_.format(Q,_,q[0:-12]+R,"%8$s"))%7$s;R=chr((39,34)[1/2>0])*12;Q=chr((39,34)[1/2==0])*3;q=%7$s|;#<?ob_end_clean();%5$seval($_=%3$s%1$s%3$s);printf($x=%3$s%2$s%3$s,$_,$x,$q,$p,$n,$s,$S,"%8$s",![]||([]+[])?$n:"");//#%4$s%6$s%7$s;print(_.format(Q,_,q[0:-12]+R,"%8$s"))%9$s',$_,$x,$q,$p,$n,$s,$S,"%",![]||([]+[])?$n:"");//#`q&%`q&%`q&%`q&%''''''''''''""";print(_.format(Q,_,q[0:-12]+R,"%"))

Validate Perl, Ruby, PHP, Python 2 and Python 3 online!

Note: running the above program in the Inspector console in Chrome (which appears to support positional arguments to console.log) returns the correct program.

Explanation

This turned out a lot longer than I'd hoped and I made my life a little harder too, so I'm likely to keep tinkering with it. I'd like to add in more languages too, but I need to find something that doesn't mind a $ sigil...

This is pretty much a standard quine format where the calculation of the quote to use is a little different for each language: in Ruby %q& is used, PHP uses ', JavaScript (Node.js) makes use of ` and Perl 5 utilises q(...). Because only the quotes change, the rest of the program is still executable and valid syntax in each language. Python 2 and 3 are wrapped around the previous program using """ for Python 2 and ''' for Python 3.

The resultant Perl and Ruby programs aren't standard quines, additional q/%qs are added each iteration, but the first program returns correctly.

The results are:

• Perl 5:
  eval($_=q&...q&);printf($x=q&...q&,...);//#'%`'%`'%`'%`''''''''''''""" ...
• Ruby:
  eval($_=%q&...%q&);printf($x=%q&...%q&,...);//#`'`'`'`'''''''''''''""" ...
• JavaScript (Chrome):
  eval($_=` ...`);printf($x=`... `,...);//#%q&'%q&'%q&'%q&'''''''''''''""" ...
• PHP:
  eval($_='...');printf($x='...',...);//#`q&%`q&%`q&%`q&%''''''''''''""" ...
• Python 2:
  ..._="""...""";...q="""...''''''''''''""";print(_.format(Q,_,q[0:-12]+R,"%"))
• Python 3:
  ..._='''...''';...q='''...""""""""""""''';print(_.format(Q,_,q[0:-12]+R,"%"))

I've re-worked this a lot, I hope I haven't missed anything key here. I still have quite a way to go to be anywhere near Jo King's score...