I didn't think that these existed but a search here told me that they did. Please will someone confirm thoughts that they don't?

Indeed, it's a quarter-tone system thing, where there are many more intervals inbetween the standard recognised semitone chromatic intervals. This is seriously not my thing, however, I'm sure someone who does use quarter tones (on instruments such like quarter tone flutes etc.) will come along and explain properly!

So yes, they do exist, but not necessarily in standard practice western music systems. How about that for a disclaimer?!

Just a thought, in addition to that above, from what I remember being taught about quarter tone systems, many moons ago!, there was also quite a bit of disagreement between different systems created on quarter tones with disparity on interval tunings and (possibly) names too. Clearly the general usage of quarter tones across the world, and historically, is not necessarily a new thing, but when fitted into the western systems of notation and analysis, by different scholars, differences can often occur. Perhaps that might be the case here too? I shall be interested myself to hear the definitive answer .