Between major and minor ...

SECONDS

If a major third divides equally in two major seconds, and a minor third divides unequally in a major and a minor second, it stands to reason that dividing the minor third equally would result in seconds between major and minor.

4:5=major third, divides equally as 8:9:10 two major seconds.
5:6=minor third, divides equally as 10:11:12 two seconds neither major nor minor.

If a scale has both a major and a minor third in the perfect fifth, the minor third will be well divided equally, but the problem is with major third.

Its first half might be 10:11, its second half 22:25. Which is close to 8:9, a little more major as an interval.

22:25=88:100
8:9=88:99

4:5-22:25=?
20:25-22:25=20:22=10:11

To make a scale without octaves or perfect forths, use a minor seventh chord, two perfect fifths (2:3) spaced 5:6 between fundamental and fundamental, as between fiftyh and fifth, and resulting in a 4:5 third between higher fundamental and lower fifth. Then take from each lower end of any third the interval 10:11, you will get another fifth divided by another minor third. The fourths with one exception will be 8:11 or 11:15, two nearly synonymous intervals:

8:11=88:121
11:15=88:120

These intervals unlike the fourth of which two add up to minor seventh and unlike tritone of which two add up to octave, add up to a major seventh, it is thus a quarter tone major than a perfect fourth and a quarter tone minor than a tritone. Obviously it sounds better than the tritone, at least the 8:11.

Lowest forth will be a bit minor than a perfect one, not by a quarter tone, but by a comma:

5:6+10:11=50:(60):66=25:33

12*11=6*22=132
6*25=150

A perfect fourth is 3:4=24:32.

24:32 > 25:33

100:110:120:132:150:165:180

If 110=330, 660, 1320 Hz="E", 100 will be a bit higher than a "D" and a bit lower than a "D sharp".

D+ - E - F+ - G - A+ - B - C+

THIRDS

If we go back to dividing the perfect fifths in fairly equal partition of major and minor third and each into equal seconds, we get the five values:

8:9:10:11:12

If major third 4:5=8:9+9:10;
If minor third 5:6=10:11+11:12;
How can I then get a third between major and minor?

By adding the greater of the two tones to the smaller of the two three-quarter-tones and the greater of the two three-quarter-tones to the smaller of the two tones.

8:9+11:12
88:99:108
88:108=44:54=22:27

9:10+10:11
90:100:110
90:110=45:55=9:11

9:11 + 22:27 = 18:22:27 = 18:27 = 2:3

4:5 = 40:50 > 44:54 > 45:55 > 50:60 = 5:6

The two intermediate thirds are thus greater than minor third, smaller than major third, nearly equal, and divide equally the fifth.

11:12=======9:10:11
===8:9:10:11:12======

This ought to be the scale that Dalecarlia, maybe Wermland and Norway too, uses so aesthetically. It has been likened to blues scale insofar that the real blue notes are not a halftone but a quarter tone below the third, fifth and seventh of major scale.

22:24:27:30:33:36:40:44

If Lyndon LaRouche and Schiller institute do not mind, let 22=220 or 440 Hz and 44=440 or 880 Hz and call both an "A". Singing like a Dalkulla requires a special technique anyway, differring from the classic one.

A - B- - C+ - D+ - E - F+ - G+ - A

Either scale is impossible on a piano, guitar, unless you tune specifically for purpose, or flute, clarinette, unless built specifically for purpose, but quite possible on violin or trombone or human voice.

The latter one - the Northern - has the distinct advantage to accomodate natural aliquotient tones.