In the past, the subject of tuning has provoked heated passion and enthusiastic debate. This is partly because music has often been regarded as more than just pleasant sound. Music was viewed as inseparable from the nature of the universe, an ideal attained in the heavens and emulated on earth. Bad music or improper tuning was a crime against nature and a danger to the fabric of society.
There was a fascination with whole numbers and the simple fractions they made. It was discovered that musical tones whose frequencies formed simple fractions like 2/1, 3/2 or 4/3 sounded especially pleasant and harmonious. Musical scales that employed these ratios were viewed as natural, a reflection of the divine order of the cosmos.
We now know that nature is far more complex than the ancient mind could have imagined. The ancient view was challenged by the Pythagorean discovery of irrational numbers. Previously, it had seemed self evident that all numbers were either whole numbers or ratios formed by whole numbers. Alas, there is an infinite quantity of irrational numbers that cannot be expressed as the ratio of two whole numbers. The ancient view of the cosmos was seriously flawed.
This "inconvenient" fact of nature would eventually influence how music was made. Purely tuned intervals based on simple ratios sound nice and so does the music based upon them, however, these simple ratios have unexpected mathematical properties that make them unsuitable for certain types of musical expression.
Purely tuned instruments have problems when they modulate from one key to another. Some keys sound good and other keys contain harsh dissonances.
What can be done to avoid this? There's a couple of possibilities. You can just live with these harsh dissonances. You can simply avoid modulations that result in these dissonant intervals. Or you could add extra purely tuned intervals to increase your freedom of modulation.
Another possibility is temperament. You can detune or temper the notes in a scale so the different keys sound either more alike or exactly alike, except for absolute pitch. Our modern system of twelve tone equal temperament allows unlimited freedom of modulation with only twelve notes to the octave. All the keys sound the same, but this system contains no purely tuned intervals, except for the octave. All the other intervals are irrational.
This is where the controversy lies. There are great practical advantages to our modern tuning system, but many people long to hear purely tuned intervals.
Some people have formed very strong opinions in favor of systems that use purely tuned intervals, perhaps viewing any sort of temperament as unnatural. Others view purely tuned intervals as impractical.
I personally don't see the need to limit myself to just one side of this question. I realise that any musical system will have some measure of tension in it because of competing musical forces. I find this to be natural and desirable. Music is dynamic and personal. Ancient theories about beauty in music may be incomplete or even wrong, but our growing understanding of music is truly exciting. This controversy about tempered verses justly tuned intervals is part of what makes music interesting.
If you make microtonal music, or are thinking of doing so, this issue is of great importance to you. You can experience for yourself the difference between tempered and purely tuned intervals and come to your own conclusions about how to express yourself musically.
I suggest you keep an open mind during your exploration. Microtonal music is full of surprises and you may find that some of your preconceived ideas will be challenged by your discoveries. I personally have gained quite a different view on tuning since I began my exploration. I have had some beliefs that I never thought of questioning until I made discoveries that caused me to think in a new way. This is part of the fun. Embrace the controversy!
See also Equal Temperament