More on synch tuning

While finding exact values for a synch tuning involves finding and
solving polynomial equations, it's easy to get an excellent
approximate answer by linearizing the logarithm. If the synch tuning
is given as 3+x, 5+x, 7+x for some small value x, while the precise
value of x is an algebraic number, it is closely approximated by

x ~= -ln(c)/(c3/3 + c5/5 + c7/7)

where c is the comma and |c2 c3 c5 c7> is the corresponding monzo.

A synch tuning can be considered better if the ratio |log2(c)/x| is
higher; some 7-limit commas with this ratio over 2 are 19683/19600,
10976/10935, 5120/5103, 65625/65536, and 4375/4374.

The x values for 10976/10935 and 5120/5103 are very close; the synch
tunings for either can be considered a slightly irregular tuning for
the temperament which combines them, hemififths:
<<2 25 13 35 15 -40||. The exact 3 for synched 5120/5103 is the
positive real root of

a^7 + 4a^6 - 1024a - 2048 = 0

Then a, a+2, a+4 are approximately 3, 5, and 7. Using this tuning, the
value of 10976/10935 shrinks to 0.24 cents, 3.7% of its true value.
2401/2400 is also a comma of hemififths; already small, it shrinks
further to this same value. If we temper a 7-limit JI scale using this
tuning, we end up with what is in effect an irregular hemififths
tuning, with two slightly different tetrad tunings, one of which is
synch beating.