Tempering stability

Tempering stability is, in a sense, the sensitivity of an interval to the tempered values of the primes relative to its simplicity. It is distinct from other harmonic senses of "stability". Very tempering-unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple similarly sized primes in it, because in any arbitrary temperament it is more likely to have an extreme detuning.

As a basic example, let's consider 11/9 vs. 15/14 (holding the octave fixed). The two have the same number of odd primes in them (11/(3*3) vs (3*5)/7) and are of similar complexity (Tenney height 6.6 vs 7.7). However, if you were to randomly detune all of the involved primes, 11/9 would be detuned on average by about 35% more. 28/27 gets detuned by over 60% more than 35/33 on average, because the former has three copies of 3 while the latter has four distinct primes, and so there are less chances for detuning to cancel out for 28/27, and detuning a single prime can have a much greater impact on the interval itself.

The most prominent unstable intervals have a factor of 9 in them; these are 9/7, 9/8, 10/9, 11/9, 13/9, and their complements. 27 and 25 also show up in more complex unstable intervals, with the interval 27/25 itself varying by 1.6 times as much as 143/105 (a stable interval with the same number of primes) and over 2.5 times as much as 35/32 (a stable interval of a similar complexity and size and with the same number of distinct primes).

The tempering-unstable intervals are an important consideration when considering the tunings of temperaments, as they are much more sensitive to detuning than the primes make apparent, and so the tunings of the primes must be carefully balanced if one wishes to preserve them; alternately they may be sacrificed to be tuned to intervals that are very different from them in just intonation, without much impacting the tuning of the primes themselves or of more stable intervals. The basic example of this is archy, where the unstable 9/8 is tuned to the stable 8/7; tunings around 27edo are common because they place the combined interval about 2/3 of the way between 9/8 and 8/7, because 8/7 is more stable and detuning it would directly affect prime 7 more than detuning 9/8 affects prime 3. However, tuning flat of 27edo moderates the detuning of the unstable interval 9/7, which is best tuned around 22edo by perfectly splitting the difference between 9/8 and 8/7. Hence, the best tunings of archy lie within those two extremes.

The instability of 11/9 also contributes to the difficulty of accurately tuning mohajira in the 11-limit. This temperament pins the tuning of the unstable 11/9 to half of 3/2; because it also flattens 3/2 by about 5 cents while tuning 11/9 nearly just, 11 itself must be flattened by about 10 cents, and only a few edos meet these criteria, lying between 24edo and 31edo or detuning 11 severely.