Other systems are meant specifically for certain uses: units representing, say, 16-tone Equal Temperament, aren't designed to compete, but to fulfill a function for that tuning only.
The following list is a concise version of the most important and interesting units. For a more comprehensive list of units, and for a pleasant converter stuffed full of ‘em, see the Tools page.
Many thanks to Manuel Op de Coul's list on Huygen-Fokker, in which he's done most of the legwork already.
1/1200 octave – Decimal: 1.000578
The now-ubiquitous measure of pitch difference suggested in 1884 by Alexander Ellis (yes, he who translated and added notes to Helmholtz), seemingly traceable to his work, 'On the Musical Scales of Various Nations'. As with most inventions, Ellis wasn't the first to come up with cents - Isaac Newton is said to have measured intervals in a similar way, as have others since.
As 12-tone Equal Temperament gained more and more dominance, and logarithms were widely operable, cents were the inevitable standard arrived at. Logically relating to a very divisible 12 (1200 per octave) and 10 (100 per semitone), cents are easy to express, understand and manipulate. Practically all broad-ranging music software uses cents in the pursuit of tuning and fine-tuning of notes.
Put simply, they're not going away in the near future.
1/180 or 1/360 octave – Decimal: 1.00386/1.00193
An approach enabling a pleasantly visual analogy, taking musical intervals as different sections of a circle.
Ptolemy is the first reported proponent of this method, choosing half the circle's circumference as an octave, allowing it to be divided into intervals of 180 degrees. Assumedly, this method is especially useful for demonstrating how commas are formed (the Pythagorean comma is about 2.5 degrees).
Later, Moritz Drobisch suggested splitting the octave into 360 degrees in a similar fashion.
1/600 octave – Decimal: 1.00116
Originally traceable to Widogast Iring in the 19th Century (a few years after cents had been suggested.) 'Irings' were justified as being close to both the Schisma and 1/12th of the Pythagorean comma. While one Iring can indeed approximate these units, theorists are generally happy to use two cents.
The name Centitone seems to have be introduced by Joseph Yasser in 'A Theory of Evolving Tonality'
John Chalmers took a cue from digitisation, creating these interesting equal scales. In 'Cents and Non-cents', Chalmers outlined 11 different tunings that split the octave into logical numbers and represented them using different numerical bases. Hexos, for example, split the octave into 1024 (2^10) and are expressed in Base-16 (hexadecimal) units. The complete 11 are:
Harmos - 1728 (12^3), b12 – Decimal: 1.00040
Hexos - 1024 (2^10), b16 – Decimal: 1.00068
Trihexos - 4096 (2^12), b6 – Decimal: 1.00017
Octos - 512 (2^9), b8 – Decimal: 1.00136
Dioctos - 1024 (2^10), b8 – Decimal: 1.00068
Suboctos - 256 (2^8), b8 – Decimal: 1.00271
Vicos - 1200, b20 – Decimal: 1.00057
Bivicos - 400, b20 – Decimal: 1.00173
Quadros - 4000, b20 – Decimal: 1.00017
Trivicos - 8000, b20 – Decimal: 1.00009
Sexos - 1200, b60 – Decimal: 1.00057
1/12th Pythagorean Comma – Decimal: 1.00114
A unit created by Werckmeister that helps to express Meantone and Well Temperaments. There are about 880 in an octave.
1/13th Tritave – Decimal: 1.00085
A measure created by Heinz Bohlen, to divide the tritave (3:1) by 13, thus aptly expressing intervals of (and deviations from) the equally-tempered Bohlen-Pierce scale.
These are convenient as the (base 10) log of 2 is 0.301. This lent them the right logarithmic favour required to understand complicated interval-scales with a linear mind.
SAVARTS (or HEPTAMERIDES)
1/301 octave – Decimal: 1.00231
1/602 octave – Decimal: 1.00115
1/30103 octave – Decimal: 1.00002
1/43 octave – Decimal: 1.01625
(301 / 7 = 43)
1/3010 octave – Decimal: 1.00023
(Variable by software)
These units are becoming ever more important as the sophistication of computer-based music making continues to rise. The last seven bits of each of two bytes are used to provide an adequate resolution, which is more than enough at a depth of 2^14 = 16,384.
The default position of any note is in the middle (8192), and usually notated in software as 0, giving a minimum of -8192 and a maximum of +8191. (Though this may easily vary depending on the software used, too.)
The minimum/maximum pitch changes (ie, how many semitones are bent down at a minimum and up at a maximum) are dictated by the destination instrument, but the recommended standard is 2 semitones each way. For reference, at this rate there would be 49,152 units per octave, though this is nowhere near possible to program using the 14 bits available.
The Tunit Calculator allows you to work them out if you know the maximum bend in 12-ET semitones of the software you're using
1/1000 octave – Decimal: 1.00069
A little more 'democratic' than the 12-ET-centric 1/1200 of cents, but a little less practical in terms of easy description of many common intervals.
1/72 octave – Decimal: 1.00967
Traced back to ancient Greek theory, this unit is favourable for its divisibility by the numbers 2, 3, 4 and their multiples.
100 of these units reside in each step of any equal temperament. Just as 12-tone ET has spawned 1200 cents, 19-tone ET would house 1900. Obviously, they're not a consistent unit, which is why they can't necessarily be used to communicate concrete ideas.
1/720 Pythagorean Comma – Decimal: 1.000018949
Introduced by John Brombaugh in order to provide a simple system for tempering. 720 can be divided by the numbers one to 10 (seven is close enough), while the syntonic comma and schisma are represented in this system by 660 and 60 respectively - again very divisible numbers, allowing tempering calculations to be performed off the cuff.
1/10600 octave – Decimal: 1.000065393
In an appropriately elaborate Ottoman way, Turkish cents are very fine units. Notice the relation of 10600 to 53, making 200 of these cents to each step of 53-ET - a good approximation of Pythagorean Tuning.
Looking through these units, you may have noticed that all fixed units can be expressed as a decimal. Indeed, this ‘unit’ is so accurate in calculating intervals that it functions as the go-between unit in the Tunit Calculator. It’s not so logically interpreted, however. Values of, say, 1.00967 for Morions, and 1.0011559 for the Centitones aren’t exactly intuitive in use, but for a ‘dumb’ computer, a decimal is the most useful unit with which to perform a unified calculation.