Comparative temperament chart
This chart shows a number of ratios and cent values for the first partials arising out of the
harmonic series and how these would relate to values produced using Pythagorean tuning on
the Gaelic harp gamut. The values of equal temperament in cents are also presented to allow
brief comparison with the modern Western European mainstream tuning standard.
Cent and ratio values are presented as though sitting within one octave.
Certain aficionados of just intonation remark that the Pythagorean major 3rd is too high in
pitch, being set at 407.82 where the harmonic series has 386.31. A similar complaint is much
more rarely heard about the major 3rd of equal temperament being 400 cents. This is nearer
the Pythagorean 3rd, and for most ordinary people it would be difficult to tell the difference
between an equal 3rd and a Pythagorean 3rd played separately since the difference of 7 or 8
cents between the equal 3rd and Pythagorean 3rd is just on the threshold of human audibility.
Roughly speaking, the Pythagorean temperament of the Gaelic harp is closer to current equal
temperament than the harmonic series and this should be no surprise, as Pythagorean
temperament nurtured the beginnings of polyphonic music in Western Christendom, the
predecessor of the harmony which equal temperament was designed to facilitate.
The values marked above in bright red are the points where Pythagorean tuning matches the
harmonic series, comprising the pitches of the first three fifths to be tuned on the Gaelic harp,
ie, G-d, d-aa & a-e.
Particularly noteworthy is the parity in pitch between the 27th partial (the first occurrence of the
major 6th in the harmonic series) and Gaelic harp tuning. This contrasts with the two options
for the major 6th provided by just temperament at 884.36 cents (5:3) and 933.1291 cents
(12:7). The minor 6th, though, is notably flatter in Gaelic harp tuning than in the harmonic
The reason for the lack in the harmonic series of a clean 4:3 ratio for the interval of the 4th is
that the harmonic series works on the power of 2, as can be seen. The 21st partial is the
closest partial to it but is flat by about 27 cents.