Semitone (minor 2nd) I: the leímma
Starting with a scale of G a b - d e - g,
tuning by 4ths produces 2 further basic whole tones of 204 cents.
5th G-d c-g
4th G-c c-f
maj 2nd c-d f-g
This produces a scale of G a b-c d e-f g. The intervals b-c and e-f are smaller than whole
tones and are called semitones (lit. half-tones) or minor 2nds, b-c and e-f being 90.225 cents
(ratio 256:243) = 294.135 - 203.91 (minor 3rd - major 2nd).
G - 204 - a - 204 - b - 90 - c - 204 d = 702
d - 204 - e - 90 - f - 204 - g - 204 -a = 702
G - 204 - a - 204 - b - 90 - c - 204 - d - 204 - e - 90 - f - 204 - g = 1200
The Greeks call this ratio of 256:243 "το λείμμα" (the remnant).
It can be seen that the difference between a fourth (498.045 cents) and two whole tones (2 x
203.91 cents = 407.82) is also 90 cents.
G - 204 - a - 204 - b - 90 - c = 498 cents
4th G-c is 498 cents
3rd G-b is 408 cents
minor 2nd b-c is 90 cents = 498 - 408
The 90.225 cents semitone, the leímma, is produced below the top pitch of a 4th when tuning
the Gaelic harp scale G a b c d e f g.
5th G-d
5th d-aa
5th aa-e
5th e-b
4th G-c
(minor 2nd b-c)
4th c-f
(minor 2nd e-f)
It is also produced above the top pitch of a 5th when tuning.
5th c-g
5th b-f#
(minor 2nd f#-g)
The difference between three whole tones, 3 x 203.91 cents = 611.73 (ratio 729:512), and a
fifth, 701.96 cents (ratio 3:2), is also 90.225 cents (ratio 256:243).
f - 204 - g - 204 - a - 204 - b - 90 - c = 702 cents