An often used definition of a microtone is one which creates an interval smaller than a semi-tone. However, for practical reasons I am going to use the term microtone to mean any tone not found in the presesnt 12 tone equal temperament scale. (In this scale every semi-tone has a frequency ratio of the twelfth root of 2.)Here is an example of the the ordinary 12 tone chromatic scale but in just-intonation; this means all frequency ratios of intervals are expressed as ratios of whole numbers; none of the semi-tones give ratios equal to the twelfth root of 2 and none are in our 12 tone equal- tempered scale! The ratios given are in relation to the first tone, C.
C: C# : D : D# : E : F : F# : G : G# : A : A# : B : C .
1: 25/24: 9/8: 22/27:5/4:4/3:45/32:3/2: 25/16 :5/3: 16/7 :15/8 : 2.
There are many possibilities to use as ratios in just-intonation. (See some of the websites listed below for this section.)
Notice that to go from one tone to another in the above scale we multiple ratios; for instance, to go a major third from G to B we compute 3/2 x 5/4 = 15/8. G to B is a major third.
To find what the interval is between two tones we divide ratios: F= 4/3, A= 5/3 so we compute
A/F= (5/3)/(4/3)= 5/3 x 3/4 = 5/4.
We get another major third. Now notice A/D= (5/3)/(9/8)= 40/27. This ratio should be 3/2 since it defines a perfect 5th like C to G. The D to A is "out of tune"; one among other reasons some people prefer equal-temperament.
Unlike traditional keyboard instruments like the acoustic piano many electronic keyboards such as the Yamaha series-the old DX7, the DX7II, SY77 or the sampling keyboards of the Ensoniq series have sub-routines or factory presets which can create a very wide variety of scales with very good resolutions for the proper frequencies.(A knowledge of logrithms and cents is often useful here.)
Sorry to say, many of the more recent virtual computer software programs like Kompackt have some factory preset microtonal scales with little or no flexibilty.
We could generalize the 12 tone chromatic scale to a 19 or 31 or "what you will" chromatic scale; this could be done in equal-temperament or in just-intonation temperament. Personally, I prefer well-tempered scales; that is, scales like the above 12 tone chromatic scale where some of the ratios are slightly altered.
Here is a terriific book about temperaments: "Tuning" by Owen H. Jorgensen, Michigan State University Press, 1991, 375 pages. It also contains a massive bibliography.