Semitone (minor 2nd) II: the apotomé
The interval of a perfect 5th, 701.96 cents (3:2), can be divided in a number of ways in Gaelic
harp temperament using whole tones of 203.91 cents (9:8), and leímmata or semitones of
90.225 cents (256:243).
3 x whole tone (major 2nd) + leímma
ie, (3 x 204) + 90 = 702
eg, f g a b c
2 x tone + leímma + tone
ie, (2 x 204) + 90 + 204 = 702
eg, g a b c d / c d e f g
tone + leímma + 2 x tone
ie, 204 + 90 + (2 x 204) = 702
eg, a b c d e / d e f g a
leímma + 3 x tone
ie, 90 + (3 x 204) = 702
eg, e f g a b
These 5ths are contain no sharp or flat notes so the semitone is always sized 90 cents.
However, a whole tone or meízon, 203.91 cents (9:8), can be divided into two semitones using
the cycle of 5ths and 4ths, and thus create a chromatic pitch between F or G (ie, F#/Gb) or
between A and B (ie, A#/Bb); but the division is not symmetrical in nature.
The leímma, 90.225 cents, is not a precise half of the meízon or whole tone, 203.91 cents.
This means that for every chromatic pitch - F#/Gb, C#/Db, etc - there will be a leímma of
90.225 cents on one side and an interval of 113.685 cents (2187:2048) on the other side,
depending on whether we use a 5th or 4th to tune the string. The Greeks call this ratio of
2187:2048 "η αποτομή" (the cutting off).
The apotomé is produced just below the highest pitch of a 5th when tuning chromatic notes.
eg, the fifth b-f# must equal 702 cents (3:2) to be perfect
b - 90 - c - 204 - d - 204 e - 90 - f - 114 -f# = 702
f-g is a whole tone (204 cents)
f-f# is an apotomé, ie, 114 cents, so f#-g ends up sized at 90 cents
eg, f# - 90 - g - 204 - a - 204 - b - 90 - c - 114 - c#
c-d is a whole tone (204 cents)
c-c# is an apotomé, ie, 114 cents, so c#-d ends up sized at 90 cents
eg, c# - 90 - d - 204 - e - 90 - f - 204 - g - 114 g#
Continuing to tune by fourths produces the apotomé just above the highest pitch of a 4th.
eg, the fourth f-bflat must equal 498 cents to be perfect
4th f - 204 - g - 204 - a - 90 -bflat (498 cents)
5th f - 204 - g - 204 - a - 204 - b - 90 -c (702 cents)
the fifth e-b is perfect and equals 702 cents
5th e - 90 - f - 204 - g - 204 - a - 204 - b (720 cents)
a-b = 204
a-bflat = 90
so bflat-b is 204 - 90 = 114
eg, bflat - 204 - c - 204 - d - 90 - eflat = 498
a - 204 - b - 90 - c - 204- d - 204 - e = 720
d-e is 204 cents
d-eb is 90 cents
eflat-e is 114 cents