Apotomé and comma
When tuning a chromatic note on the harp such as G#/Ab, Pythagorean tuning therefore
offers two options for tuning, the two semitones of different sizes called the leímma and
apotomé. This creates a choice of two possible pitches for the chromatic note. Tuning up by
a 5th from a C# to the chromatic note between G and A will produce an apotomé between the
G and the chromatic note but a leímma between the chromatic note and the A. Tuning up by a
4th from an Eb to the chromatic note between G & A will produce a leímma between the G and
the chromatic note, and an apotomé between the chromatic note and the A. Therefore, tuning
up by a 5th will raise the pitch of the chromatic note higher than tuning up by a 4th.
Represented in the diagram above by an orange box is the part of the apotomé lying between
the two chromatic pitches produced by the different tunings, an interval of 23.46 cents
(531441:524288, roughly 80:81). It is called by the Greeks "το Πυθαγόρειο κόμμα" (the
Pythagorean 'comma', lit. piece). From this, it can be seen that the sharp pitches of
Pythagorean Gaelic harp tuning, eg, f# & cc#, are pushed by the apotomé so close to the
upper pitch that they can be called gflat & ddflat.
The apotomé, at 114 cents (2187:2048), is the size of minor 2nd nearest to the 17th partial,
which is the first minor 2nd to occur in the harmonic series, 17:16 (105 cents). It is therefore
noteworthy that it is not the size most favoured in Gaelic harp tuning and that the preference is
for the smaller leímma, 90 cents (256:243), found in the natural octave G a b-c d e-f g, at b-c
and e-f.
This preference is evident from the fact that only the following semitones are produced by the
Gaelic harp tunings specified by Edward Bunting and John Bell and all would be sized 90 cents.
e-f, b-c, f#-g, cc#-dd
Whereas leímmata will be heard between two strings tuned to f#-g or cc#-dd, apotomaí,
obviously, cannot be played at f-f# or c-c#, since a natural and a sharp cannot be produced by
the same string during continuous playing as the Gaelic harp has no levers or pedals.
* tuned to a 5th from C#
* tuned to a 4th from Eb