31

By adding a relative 7-goodness measure for the n-et to the same for
the m-et, where n+m=31, I got the following ordering of white-keys,
black-keys, which is probably as good a one as any to consider them
in: 27+4, 19+12, 22+9, 16+15, 26+5, 21+10, 29+2, 25+6, 24+7, 18+13,
30+1, 28+3, 17+14, 20+11, 23+8.

27+4

From the continued fraction for 4/27 we get 1/7 as an approximation,
and 1/4 < 8/31 < 7/27 locates our generator. It is a *flat* minor
third, though a bit better than that of the 12-et as we can see from
the above inequalities, so this is not the same system as the 8/19
system, but is closely allied to 7/27 and 15/58. If 5/19 and 9/34 are
called kleismic then this one shouldn't be; in any case the kleisma
isn't very relevant. We have the following table:

3 10 10
5 9 10
7 7 10

The first column gives the prime, the second the generator steps to
reach it, and the third the p-complexity, which can be seen from the
generator steps and is in this case 10 in all three limits--as could
have been predicted from the way I sorted these, this is a good
system for the 7-limit. We have a 7-note scale with pattern 1717177,
an 11-note scale with pattern 11611611616 and a 15-note scale with
pattern 111511151115115. While we have a complexity of 10, this does
not mean the 7-note scale lacks for harmonic resources; from
10-9=1 we get many major sixths and minor thirds, from 10-7 = 3 many
septimal major sixths and minor thirds, and from 9-7=2 septimal
tritones. The significant commas for this system are the small
septimal comma of 126/125, the breedsma of 2401/2400, and their ratio
(126/125)/(2401/2400) = 1728/1715, which doesn't seem to have
acquired a name.

19+12

This is, of course, 1/4.15-comma meantone, and 7, 12 or 19 meantone
fifths will give what can be considered scales or temperaments, as
you choose. While the tendency to think this is the only way to
arrange the notes of the 31-et should be resisted, it is both obvious
and excellent for 5-limit harmony, and does well beyond it. From
Euclid we get the approximation 5/8 for 12/19, and locate the
meantone fourth by 5/12 < 13/31 < 8/19. If we look at the Farey
sequence in this neighborhood, up to denominators of 75, we get
5/12<28/67<23/55<18/43<31/74<13/31<
21/50<29/69<8/19<27/64<19/45<30/71<11/26,
which pretty well defines the range of the meantone (if we go lower
we get schismic instead, and 30/71 already has better fifths and
thirds than the meantone is giving it.) While it hardly seems
necessary to point it out, the diatonic scale with pattern 5535553
and the 12-meantone temperament of 332323323232 fall out by the usual
methods. With a 3-complexity of 1, 5 of 4, 7 of 10 and 9 of 10, this
is an excellent system and deservedly popular.

22+9

After approximating 22/9 by 5/2, we locate the generator by
2/9 < 7/31 < 5/22. If we look at the Farey sequence up to 90 from
7/31 to 5/22, we see that this defines a certain family:
7/31<19/84<12/53<17/75<5/22, whose generator is a sharp septimal
minor third (7/6); if no one has yet named it I propose the George
Orwell, because of the 19/84 generator. We get the following
orwellian values for generator steps:

3:7 5:-3 7:8 11:2

This gives a 5-complexity of 10, 7 of 11, 9 of 17 and 11 of 17; the
31-et also has 13 and 17 of 17. The system abounds in intervals of
12/7 and 7/6 (of course), as well as 11/8, 5/4, 11/5 and 11/6. The
9-note scale has pattern 434343433, and we also have a 13-note scale
of pattern 1331331331333.

16+15

16/15 is of course approximately 1/1, and 1/16 < 2/31 < 1/15 defines
the location of this generator. It belongs to a small family,
2/31 < 5/77 < 3/46, which all have 3: 9, 5: 5, 7: -3, and 11: 7 as
the number of generator steps to reach the 11-limit. This gives
complexities of 9 for 3, 9 for 5, 12 for 7, and 21 for 9 and 11.
Characteristic intervals are 11/5, 11/6 and 5/3. The 15 and 16 note
scales are of course respectively 2x14 3, and 2x15 1.

26+5

The good times just keep rolling with the 31-et; 26/5 is of course
approximated by 5, and we have 5/26 < 6/31 < 1/5, as well as another
small family, 5/26 < 11/57 < 6/31, all of which have
3:3 5:12 7:-1 11:-8 13:14 for generator steps. This system features a
generator of 8/7, which is too nearly exact in the 26-et to give a
very good value for the fifth. Characteristic intervals are 7/4 (of
course), 3/2, 7/6, 9/7, 9/5, 13/10 and 11/7. We have a 5-note scale,
66667, and an 11-note scale, 51515151511.

21+10

This is the first glimmer of the miracle system, unless you want to
count the 10-et. From 21/10 we get 2/1, and hence 2/21 < 3/31 < 1/10.
If we look at the Farey sequence for denominators less than 100 after
3/31, we get 3/31 < 7/72 < 4/41, and this defines the miracle family.
The 10-note scale with pattern 3333333334 is the most obvious choice,
but there's also 11 notes and 33333333331. Characteristic intervals
are 7/4, 3/2, 5/4, 7/5, 11/9 and 11/6.