Friday, August 1, 2008

Getting the most from your hand-held tuner.

1. Understanding Scales and Intervals.



An "interval" is the ratio of the frequencies of two notes in a scale. Let me give you an example. The interval between Low A and B in our scale is 9/8 (or 1.125). For instance, a reasonable frequency for our Low A would be 470 Hertz (or 470 vibrations per second). In a correctly set chanter with this Low A then the "B" must be 1.125 times 470 which equals 528.75 Hertz.

It is important to realise that the actual frequency of Low A can be fixed anywhere between say 460 and 480 Hertz - this will depend on many things including the make of chanter, the mood of the pipe major, and the weather !!! The point to remember is that the interval between Low A and B will always be the same - i.e. 1.125

At the risk of labouring the point, the "intervals" between notes in the bagpipe scale are fixed and immutable, even though the actual frequencies may vary over a limited range.

2. The McNeill/Linehan Scale.

The bagpipe scale is unique, and the recognised intervals, (as reported by McNeill and Linehan in “Acustica” vol 4, 1954) are as follows :

Low A to B 1.125 (9/8)
Low A to C 1.25 (5/4)
Low A to D 1.35 (27/20)
Low A to E 1.5 (3/2)
Low A to F 1.666 (5/3)
Low A to G' 1.8 (9/5)
Low A to A' 2 (2/1)


It is worth noting that, at that time, the Low A was deemed to be 459 Hertz.


Summarising the McNeill/Linehan Scale







Chanter A B C D E F G A
Tuner A# C D D# F G G# A#
Correction(cents) 0 +4 -14 +20 +2 -16 +17 0
Freqency Hz 470 528.7 587.5 634.5 705 783 846 940

To fully understand the meaning of “Cents” you should refer to the section 5 below.

However, the bagpipe scale has shifted quite dramatically from this early standard, especially since the early 1970’s. The accepted pitch has moved upwards quite markedly to be in the “470+ Hertz” region, while D and G have been considerably flattened. We will now look specifically at the Modern Scale.

3. The Modern Bagpipe Scale.

As previously mentioned, the trend since the 1970’s has been towards higher pitch and also towards a fairly large flattening of D and G. The correction for D should be +2 cents (rather than +20) and the correction for G should be -24 (rather than +17). The other notes are unchanged. This will place the scale in the range of a number of top solo players and bands of the past 30 years and are as follows :


Low A to B 1.125 (9/8)
Low A to C 1.25 (5/4)
Low A to D 1.333 (4/3)
Low A to E 1.5 (3/2)
Low A to F 1.666 (5/3)
Low A to G' 1.75 (7/4)
Low A to A' 2 (2/1)

The Modern scale will therefore require the following corrections on the tuner.
(n.b. the corrections are calculated for the nominal ratios listed immediately above.)






Chanter A B C D E F G A
Tuner A# C D D# F G G# A#
Correction 0 +4 -14 -2 +2 -16 -31 0
Freq Hz 470 528.7 587.5 626.6 705 783.3 822.5 940

Notice that the ratio of Low A to High A is 2. In other words High A is precisely twice the frequency of Low A. This "two-to-one" ratio is called an "Octave".

For the record, the tenor drone is precisely one octave below Low A, and the bass drone is precisely one octave below the tenor drone. Also, Low G is an octave below High G.

4. Where are the drones in all of this.

If we assume a Low A frequency of 470 Hertz, the tenor should be set to 235 Hz, and the bass to 117.5 Hz. The relationship of particular chanter notes with the drones can therefore be expressed as follows.

LowA is twice the tenor frequency and 4 times the bass frequency) = 470Hz.
C is 5 times the bass frequency = 587.5Hz.
E is 6 times the bass and 3 times the tenor frequency = 705Hz.
HiA is 8 times the bass and 4 times the tenor frequency = 940Hz.


Those notes which have audible relationships with the drones (A, C, E and HiA) are pitched identically in both the McNeill and Modern scales while D and G are significantly different.


Firstly, we need to establish how far each interval of our two scales deviates from particular intervals in the Equal Temperament or Chromatic Scale used by the normal hand-held tuner. If you intend to use your electronic tuner to set a chanter (as opposed to its normal function of tuning drones) you need to know the precise deviation that will be displayed for each chanter note on the tuner.


To fully understand the rationale behind all this, it will be necessary to digress briefly into a little more basic music theory.

5. Equal Temperament - Semi-Tones and Cents.

In the "Equal Temperament" approach the octave is divided into 12 equal intervals and each is called a "semitone".

To further complicate matters, each of the 12 equal intervals (or semitones) is given a value of 100 cents, the whole octave then being represented by 1200 cents. The importance of this for pipers is that the deviations are expressed in the form of "cents".

The semitone is a precise interval and is 1.0595. (For the mathematically minded, this is the twelfth root of 2).

Of course there are only eight notes in an octave, therefore not all of the intervals between notes are semitones. In fact some are "tones" which have intervals of 1.1225 (which is 1.0595 squared) and these have a value of 200 cents.

To recap:

a semitone = 100 cents and is an interval of 1.0595.
a tone = 200 cents and is an interval of 1.1225.
an octave = 1200 cents and is an interval of 2.

Scale intervals referred to a chosen note (in our case "Low A") can be expressed as Cents using the following formula:

Cents = (1200 x log(interval))/ log 2
= 3986 x log(interval)


To illustrate this by an example let us choose "F" on the bagpipe scale. Referred to Low A, the interval is 1.666 (as noted earlier).

Cents = 3986 x log(1.666);
= 3986 x 0.2218
= 884 cents

We will discover shortly that this is quite close (within 18 cents) to the interval between A# and G in the Equal Temperament scale.

The Low A of the Pipe chanter is closest to A# (B flat) in the chromatic scale, therefore we should look at the intervals referred to this note to see what is closest for each of the other chanter notes.

The relevant intervals for the Equal Tempered Scale referred to A# can be tabulated (in Cents) as follows:





A# C D D# F G G# A#
0 200 400 500 700 900 1000 1200

The intervals for the Modern bagpipe scale can be tabulated (in Cents) as follows:





A B C D E F G A
0 204 386 498 702 884 986 1200

The differences (in Cents) between the nearest comparable notes (chanter and tuner) are therefore:

0 +4 -14 +20 +2 -16 -31 0

6. Summarising - the Modern Scale.








Chanter A B C D E F G A
Tuner A# C D D# F G G# A#
Correction 0 +4 -14 -2 +2 -16 -31 0
Freq Hz 470 528.7587.5626.6705783.3822.5940





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If you are not now totally confused then you haven't been paying attention !!!!!


Old Angus




3 comments:

Angus Jn said...

Two points to add .. first where did we get get the modern scale from? Well a few years ago we gathered up recordings of the winning bands at the Worlds for a few years, and recordings of the top solo pipers and measured where they were placing the notes. The above relationships came from analysing these recordings.

Also its interesting to look at whats happened to the high G ... since its moved to align with the 7th harmonic of the bass drone giving as a pure scale based on the bass
ie Bass * 4 = A
Bass * 5 = C
Bass * 6 = E
Bass * 7 = High G
Bass * 8 = High A

and then we have low G, B, D and F as the notes that dont blend with the drones. Changing a tune from one of these keys to the other can alter its mood considerably.

Also as a side issue -- the movement of D and G are important when playing seconds since old books (say anything written before 1980) were assuming the Seumas McNeill scale and those seconds simply dont work any more

Angus

Anonymous said...

I would be surprised if the MacCrimmons knew any of this stuff !

Donald Mor said...

Actually, Anon, we knew a lot more than you realise.