© Ewan Macpherson, 1998

Winter 1998 issue of the magazine

While the pitches of the drones and the tonic note on the chanter are
referred to as 'A', they are actually much sharper (higher than
B^{b}) than this on the modern Great Highland Bagpipe. Presumably
the pitch was close to A when the music first began to be written down,
but over the course of the past century, pipe makers have been producing
sharper and sharper chanters. The standard for the frequency of the A
above middle-C is 440 Hz (cycles per second), although modern concert
bands and orchestras tend to tune a little sharper than this. The
standard frequency of B^{b} is higher by a factor of the
12^{th} root of 2 (1.05946), i.e. 466 Hz. Published measurements
from 1885 show the pitch of one chanter to be 441 Hz, while the average
of several chanters was found to be 459 Hz in the mid 1950s. Recent
measurements show that modern chanters tend to tune between 470 and
480 Hz for low A. In most of the following discussion, I will
ignore this pitch discrepancy and refer to the notes by their "chanter
names".

Pipers think of the scale of the chanter as consisting of the notes in
the octave between low A and high A with an extra low G below low A. The
notes are named low G, low A, B, C, D, E, F, high G, and high A.
However, if one plays these notes on a piano, one will not hear a scale
resembling the chanter scale, even if one transposes up to
B^{b}. This is because the sizes of the steps between some notes
are incorrect. The note we call 'C' is really closer to C#, and the note
named 'F' is really closer to F#. Since traditional pipe music doesn't
use non-sharp C and F, we don't bother to indicate the sharp signs, but
our music could be written with a D key-signature (containing C# &
F#) to avoid confusing non-pipers. So what scale __is__ A, B, C#, D,
E, F#, G, A ? If the G were G#, then this would be an A Major scale,
however the chanter G is close to G-natural. One can describe this scale
as "a major scale with a flattened 7^{th} or leading tone". This
is also known as the Mixolydian mode. The same scale steps can also be
obtained on the piano by starting on G and playing all the white keys up
to the next G.

While the scale is indeed a major scale with a flattened 7^{th},
the precise frequencies of the notes on the chanter do not match those
of a piano, guitar, or standard Western wind instrument playing that
scale. Western instruments are tuned in "equal temperament" (ET), which
means that the frequency ratio between any two notes a semi-tone apart
is intended to be the aforementioned 12^{th} root of 2
(1.05946). While this means that intervals between notes in chords
generally do not match exact harmonic ratios, it also means that these
instruments will be equally in (or out) of tune in any key. Western
Classical music makes extensive use of key modulations, so this is a
desirable property. 4^{th}s (like from A to D) and
5^{th}s (from A to E) are well-represented in this scheme, but
major 3^{rd}s (A to C#) and 6^{th}s (A to F#) are
actually quite far from exact harmonic ratios.

Pipe music does not require this ability to modulate to different keys. The nine notes available on the chanter are fixed, the chanter melody is played against the unchanging tone of the drones, and traditional pipe music does not involve harmony-playing between chanters. Thus, we are able to trade in harmonic flexibility for much enhanced purity of tuning.

From the time of Pythogoras, it has been known that two notes sounding
together are heard as a consonance (a smooth, pleasant combination) when
the ratio of their frequencies is a fraction with small integers in the
numerator and denominator. In such a situation the two notes have many
harmonics in common, which reduces the roughness of the combination. A
simple example is the octave (from low A to high A) in which the
frequency ratio is 2:1 and every harmonic of the upper note coincides
with a harmonic of the lower one. A ratio of 5:4 gives a major
3^{rd} (A to C#), and a perfect 5^{th} (A to E) has a
ratio of 3:2. In solo pipe music the frequency ratio we are concerned
with is always that between the chanter and the drones. Since the drone
notes are "A" (one or two octaves below the chanter's low A) we can just
define the frequency of each note on the chanter by the ratio of its
frequency to that of low A. Thus for example, low A will have a ratio of
1:1, and high A will have a ratio of 2:1.

Not all of the frequency ratios on the chanter are universally agreed-upon, and tuning styles, like the overall pitch, have changed with time. Some notes (the As, C, and E for example) are very easy to set relative to the drones, with clearly audible "beating" when out of tune, but others are less straightforward. In particular there are several schools of thought on the tuning of D and high G. In the non-piping world, the simplest and purest tuning scheme known for a Mixolydian scale is a type of Just Intonation (which simply means a scale using whole-number frequency ratios) in which D is tuned to a frequency ratio of 4:3 above low A, and the high G is tuned to a ratio of 16:9. However, in 1954, J. Lenihan and Seumas MacNeill published a study of the tuning of 18 pipe chanters in which they concluded that Ds were tuned to 27:20, and the high G to 9:5. These values are sharper than the simple Just values, although since they are whole-number ratios they are, strictly speaking, just. A scale formed in this way has unique and interesting melodic properties, particularly when one considers the various pentatonic scales which can be derived from it (Piping Times, October 1997).

Analyses of recent recordings by well-known soloists from North America
and the U.K. show that tuning practice has changed in the last 4
decades. Ds vary from piper to piper, but are generlly tuned very close
to the Just D (a 4:3 ratio with low A) rather than the significantly
sharper 27:20 ratio. High Gs are also tuned much flatter, apparently at
a ratio of 7:4, which might be termed the "Harmonic" high G since in
this case the note's harmonics coincide exactly with every
7^{th} harmonic of the bass drone. Low Gs are frequently exactly
an octave lower at 7:8. In practice, the tuning of high A almost always
departs from a true octave relationship with the drones, and is typially
tuned 10 to 30 cents flat. Recordings of John D. Burgess provide good
example of this effect. It is perhaps done in order to make the high A
more audible against the drones or to guard against the reed sharpening
up during a performance, but is probably best regarded as a matter of
convention or taste. Certainly the ear of the pipe music afficionado
becomes used to the flat high A to such an extent that a "true", octave
high A can sound overly sharp and shrill.

Before finally getting to a listing the frequencies of the notes of
the chanter scale, it will be useful to describe a second method of
indicating frequency ratios. The interval of a semitone (for example,
from C(#) to D on the chanter) is a frequency difference of about 6%. In
the range of frequencies covered by the pipe chanter, the human ear can
detect frequency changes of about 0.1%, so using the semitone to
describe fine tuning is clearly too coarse a measure. For this reason
tuning theorists use a unit called a cent. A cent is a frequency ratio
1/100^{th} of a semitone in size. In other words, a cent is a
ratio equal to the 1200^{th} root of 2, which is approximately
1.00058. Most electronic instrument tuning devices give a readout by
indicating the closest standard pitch to the one being measured along
with the deviation in cents from this frequency.

To calculate the size in cents of the ratio between frequencies
f_{a} and f_{b}, one of the following formulae can be
used, depending on your favorite type of logarithm:

c = 1200 × log_{2}(f_{b}/f_{a}) or 1200 ×
log_{10}(f_{b}/f_{a}) / log_{10}(2) or 1200
× ln(f_{b}/f_{a}) / ln(2)

For example, the ratio from 200 Hz to 300 Hz (3:2) is

1200 × ln(300/200)/ln(2) = 701.96 cents.

This interval is a perfect 5^{th}, which in ET is 7 semitones, or
700 cents. Thus the ET interval is only about 2 cents off from a pure
5^{th}.

Finally, here is a table indicating the frequency ratios between the
notes of the chanter and low A. The "Closest ET Note" column shows the
closest note *disregarding the pitch discrepancy discussed
previously*. For a closer approximation to the true scale, these
notes should be transposed up by a semitone (A > Bb, etc.). The
"Just", "MacNeill", and "Harmonic" Ds and Gs are indicated by
D_{J}, D_{M}, G_{J}, G_{M}, and
G_{H}. Note that the MacNeill high G is not an exact octave
above either of the low Gs.

Note name | Ratio to low A |
Cents from low A |
Closest ET note |
ET cents | Deviation from ET |
Freq for A = 466 Hz |
Freq for A = 475 Hz |

high A | 2:1 | 1200.0 | A | 1200 | 0.0 | 932 | 950 |

high G_{M} |
9:5 | 1017.6 | G | 1000 | +17.6 | 839 | 855 |

high G_{J} |
16:9 | 996.1 | G | 1000 | -3.9 | 828 | 844 |

high G_{H} |
7:4 | 968.8 | G | 1000 | -31.2 | 816 | 831 |

F(#) | 5:3 | 884.4 | F# | 900 | -15.6 | 777 | 792 |

E | 3:2 | 702.0 | E | 700 | +2.0 | 699 | 713 |

D_{M} |
27:20 | 519.6 | D | 500 | +19.6 | 629 | 641 |

D_{J} |
4:3 | 498.0 | D | 500 | -2.0 | 621 | 633 |

C(#) | 5:4 | 386.3 | C# | 400 | -13.7 | 583 | 594 |

B | 9:8 | 203.9 | B | 200 | +3.9 | 524 | 534 |

low A | 1:1 | 0.0 | A | 0 | 0.0 | 466 | 475 |

low G_{M} |
8:9 | -203.9 | G | -200 | -3.9 | 414 | 422 |

low G_{H} |
7:8 | -231.2 | G | -200 | -31.2 | 408 | 416 |

Further reading

The study of musical scales and tunings has a long and distinguished history
in mathematics, acoustics and perceptual psychology, and any good textbook
on musical acoustics or the psychology of music will address this subject
and give pointers to more material. For information on Seumas MacNeill's
study of the Highland bagpipe scale see the book *Piobaireachd and its
Interpretation* by Seumas MacNeill and Frank Richardson, and the article,
"The Great Highland Bagpipe Scale" by D.M.B., in Piping Times Vol.50, No.1,
October 1997.