Why the comma exists
Multiplying or dividing is an infinite process. However, dividing by 2 produces finite individual
values whereas dividing by 3 produces infinite individual values. These sets of values cannot
match except at a value of 1.
The harmonic series is a lattice of ratios, such as 1:1, 2:1, 3:1, 4:1, 5:1, 6:1, 7:1, 8:1, 9:1 and
so on, which produce a series of ascending intervals. The intervals are produced by one
fundamental pitch, eg, on one harp string.
Within this lattice are different sets of ratios, some of which do not match. The set running 1:1,
2:1, 4:1, 8:1, 16:1 etc produces a series of consecutive octaves ascending in pitch. The set
running 1:1, 3:1, 9:1, 27:1 etc produces a series of consecutive fifths ascending in pitch. A set
of ratios running 4:3, 16:3, 64:3 and so on, produces a series of consecutively ascending
fourths.
Tuning up in octaves (2:1) is like dividing by 2.
G - g - gg - ggg - gggg - ggggg - gggggg - ggggggg etc
Tuning up in fifths (3:2) is like dividing by 3. Tuning up in fifths never produces a pitch that
sounds exactly like an octave. This is because values created by dividing by 2 cannot match
values created by dividing by 3, as illustrated above, and this proceeds infinitely. The closest
interval to an octave in tuning a series of consecutive fifths would be the interval equivalent to
7 octaves plus a Pythagorean comma which is about 23.46 cents (531441:524288, roughly
80:81). This would sound like a sharp octave.
G - d - aa - ee - bb - ff# - ccc #- ggg #- dddd# - aaaaa# - eeeee# - bbbbbb# - ffffff##
To keep the matter brief, tuning upwards in 8ves & 5ths & 4ths uses three separate sets of
ratios. Apart from the fundamental, any pitches produced in one of these sets do not perfectly
match any pitches of the other sets.
The 5th and 4th added together make a perfect octave and can be seen as invertions of each
other. Tuning up by a 5th of 701.955 cents (3:2) or tuning down by a 4th of -701.955 cents
(2:3) from the 8ve produces a central note of similar value (eg, one major 7th of a single ratio).
5ths G - d - aa - ee - bbb - fff#
|
(243:128) G-f# of 1109.77 cents
|
Inverted 4ths F# - b - e - aa - dd - gg
However, whenever tuning up by 5ths and also tuning up by 4ths towards the same pitch, the
symmetry of invertion is lost and ratios will not match because two separate sets of ratios are
being used which ascend side by side, as it were. In this new case, the nearest thing to an
octave created is two notes of distinct ratios.
5ths G - d - aa - ee - bbb - fff# (243:128) 1109.77 cents
4ths G - c - f - bbflat - eeflat - aaaflat - ddddflat - gggflat (4096:2187) 1086.315 cents
The difference of interval between sharp and flat in the above example is one Pythagorean
comma, roughly 23.46 cents (80:81). The comma occurs since the two set of ratios cannot
produce matching values because they are not inverted series but separate series of
consecutive intervals ascending side by side. The comma is the nearest to a unison or octave
that can be reached when tuning up in 5ths and 4ths. The following is an another example.
5ths G - d - aa - ee - bbb (81:64) 407.82 cents
4ths G - c - f - bbflat - eeflat - aaaflat - ddddflat - gggflat - ccccflat (8192:6561) 384.36 cents
A similar effect is, naturally, produced when tuning down by 5ths and up by 4ths. As the rising
series of 5ths and 4ths are fully chromatic in nature, and since the individual has a choice of
tuning up or down in either 5ths or 4ths, the phenomenon of the comma between the note
pairs naturally explains why Pythagorean tuning offers a choice of two sizes of semitone.
With regard to the practicalities of tuning up on the Gaelic harp, it can be seen that if one
tuned perfectly as follows (up the 4th and down the 5th, which equals tuning up in 4ths):-
cc bbflat aaflat
/ \ / \ / \
g f eflat dflat
then the pitch would not be exactly the same as that produced by tuning up perfectly as follows
up the 5th and down the 4th, which equals tuning up in 5ths):-
d e f#
/ \ / \ / \
G a b c#
However, such considerations will not affect those who play the traditional diatonic Gaelic harp
repertoire and tune their Gaelic harp in accordance with historical data. It is sufficient to note
that traditional tuning allows the comma to affect chromatic notes in such a way that all major
3rds on the instrument are preserved at the same interval of 407.82 cents (two meízones or
whole tones, at G-B, A-C#, C-E, D-F# & F-A) and all minor 3rds kept to 294.135 cents (a
meízon and a leímma, at A-C, B-D, C#-E, D-F, E-G, F#-A).
etc
0.125 (8:1)
0.25 (4:1)
0.5 (2:1)
1 (1:1)
etc
0.037037... (27:1)
0.111111... (9:1)
0.333333... (3:1)
1 (1:1)