Tuning tables

After giving modern equal temperament for comparison, this page details four tunings that could be used on an early Gaelic harp:
medieval pythagorean tuning with pure 5ths;
Renaissance meantone temperament with pure 3rds;
Highland bagpipe just intonation;
and my own just intonation.

Equal temperament

For quick reference, modern equal temperament is given first.

Every tone is 200 cents and every semitone is 100 cents. None of the intervals are consonant ratios.

The Harmonic Series

The natural harmonc series is also relevant since these notes are naturally present as “overtones” in any normal note. These are also the pitches played by natural trumpets and horns. This chart gives the harmonic series from the 8th through to the 15th harmonic, producing the “natural” diatonic scale. While I don't consider this a useful tuning for a harp, it gives some context for features of other tunings.

Pythagorean

The Pythagorean intonation tunes all 5ths pure, which means that the 3rds are wide and sour.

All of the 5ths are 3/2 (702 cents), and all of the 4ths are 4/3 (498 cents). This means that any movement in parallel 5ths or 4ths is totally consonant and beautiful. However, all the 3rds are very dissonant at 81/64 (408 cents) and all the minor thirds are dissonant at 32/27 (294 cents). Similarly, all the 6ths are dissonant at 27/16 (906 cents).

Pythagorean tuning dates back to ancient Greece, and was used in medieval European music a lot. At least, medieval scholars specified it as the only permitted and consonant type of tuning. Because of the medieval origins of the Gaelic harp tradition, and its conservative nature, this is the most obvious tuning to use. The harmonic basis of Gaelic harp music right to the end has been demonstrated to be based mostly on movement in 5ths and octaves⁵, which would also support this tuning method.

Pythagorean tuning is even, in that every similar interval is identically sized*. Every 3rd is 81/64, every 6th is 27/16, every 5th is 3/2.

Every tone is 9/8 (204 cents) and every diatonic semitone is 256/243 (90 cents)

Pythagorean tuning is easy to tune on a harp, using a cycle of 5ths and/or 4ths. Every 5th and 4th is made absolutely pure. Edward Bunting's tuning sequences are the obvious sequences to use - for more details see my tutor book tuning page.

Mean tone

The 1/4 comma meantone temperament tunes all major 3rds pure, which means that all other intervals are out-of-tune. To get every single major 3rd pure, the 5ths are 'tempered', that is each 5th is narrowed by a small amount. This is a temperament not a tuning, since the ratios are no longer simple numbers.

As you can see, the only pure consonant intervals are the major thirds. The minor 3rds, 4ths, 5ths and 6ths are all tweaked out of tune to allow this.

Meantone temperament dates from the early 16th century and is used for musical styles where there are a lot of triad chords, where the sweetness of the pure 3rds is desired. Meantone tuning was probably not used by Gaelic harpers for playing their native repertory. If it was used at all, then it would be for harpers playing continental style polyphonic music, or working in consort with other instruments such as lutes and viols.

Meantone tuning is even, in that every similar interval is the same*. Every tone is 193 cents and every diatonic semitone is 117 cents. The tempering process produces irrational ratios.

1/4 comma meantone is a little tricky to tune on the harp. You need to learn how to judge a slightly out of tune 5th, slightly narrow, so that when you do four of these consecutively (G-D, D-A, A down an ocatve, A-E, E-B, B down an octave) your resulting 3rd is absolutely pure. It is tricky because each 5th has to be the same amount out-of-tune. Once you have got that far, either tune the other strings by pure 3rds, or continue the cycle of out-of-tune 5ths. Damp the other strings, especially na comhluighe and cronan Gs, so their sympathetic vibrations don't impair your judgement of the intervals.

Just intonation

There are many different just scales. The idea of a just scale is that there is more than one size of tone: a major tone of 9:8 (204 cents) and a minor tone of 10:9 (182 cents). Together these combine to make a major third of 5:4 (386 cents). You can see that the harmonic series (above) has these intervals.

To make a minor third of 6/5 (316 cents), you can combine the major tone with a minor semitone of 16:15 (112 cents), or you can combine the minor tone with a major semitone of 27/25 (133 cents).

Now, arranging these major and minor tones and semitones in different sequences gives subtly different scales, with the pure and the impure 3rds, 4ths and 5ths in different places.

Bagpipe scale

This tuning for Great Highland Bagpipe was published by Seamus MacNeill in 1968⁶.

(in actual fact, the pipes have two more notes than this. High A is an octave above low A, or 10/9 above the flat 7th G. There is also a low G, which is 9/8 below the low A, and so does not make an octave with the high G. But we can still discuss the scale shown here without worrying about these peculiarities).

For ease of comparison, I have transposed this scale to G, and will use this transposed version in the discussion below.

Although the drone sounds continuously on transposed G, and the pipes are unable to play any other simultaneous pairs of notes, this scale is interesting in that it is weighted towards consonant 3rds. Every minor 3rd is consonant at 6/5 (316 cents), and every major 3rd is consonant at 5/4 (386 cents) except the transposed C-E which is wide at 100/81.

Of the six 5ths, four are consonant at 3/2 (702 cents) while two are narrow and dissonant at 40/27 (681 cents) (transposed A-E and C-G). Similarly with their matching 4ths. The curious thing about this scale is that these consonant pure 3rds are only audible as melodic intervals, since they can't be sounded simultaneously. Instead, the intervals that can and are sounded simultaneously (that is, every note singly against the drone on transposed G) are given some of them pure consonant intervals, and others slightly dissonant ones. This gives the pipe scale a lot of character. The 3rd C♯ (transposed B) is very pure and sweet against the drones, while the 4th D (transposed C) is unstable and wide (sharp) agaist the drone. The 5th and 6th, E and F♯ are pure and consonant with the drone, while the flat 7th is sharper than usual at a very consonant 9/5, giving the characteristic 'mixolydian' scale of the pipe an extra beauty.

This bagpipe scale is uneven; it has two different tones. The major tone is 9/8 (204 cents) and the minor tone is 10/9 (182 cents). The semitone is 27/25 (133 cents).

Barnaby Brown has experimented with this tuning on a harp (though he uses a non-traditional tuning sequence), and he has published some notes and a demonstration on his blog. See also his article in Piping Today 38, 2009.

My just intonation

This just scale is perhaps closer to what I have been using on the harp for some years. It was only after discussing the subject with Siobhán Armstrong in August 2009⁷ that I realised that this was a possible model for what I was doing; that conversation led to this analysis of different scales. (I have adjusted this scale’s flat 7th, Oct 2013).

You'll notice that this scale is very similar to the pipe scale above; only the c is different, becoming pure against the G drone of the harp. This just intonation makes four out of the six 5ths consonant and pure (3/2 or 702 cents); the sour 5ths are F to C and A-E (40/27 or 680 cents). This allows some of the 3rds to be pure 5/4 (386 cents) (G-B, C-E, F-A) or 6/5 (316 cents) (B-D, E-G, D-F). There is a single dissonant wide pythagorean 32/27 (294 cents) (A-C).

When the F is retuned to F♯, the F♯ ends up a pure 5/4 major 3rd above D and a pure 6/5 minor 3rd below A.

This scale is an uneven tuning, with two different sized tones: a major tone of 9/8 (204 cents) and a minor tone of 10/9 (182 cents). The diatonic semitone is 16/15 (112 cents).

I am tuning this scale by following the cycle of 5ths sequence in Bunting's notebook. The 'just' E is tuned to the A below as if to set a pythagorean 5th, but the E is tuned to sound pure agains the implicit drone of the sympathetic resonance of the whole harp, instead of just against the struck A in isolation. Then the cycle of pure fifths continues E-B.

New for Oct 2013: I have made instructions for tuning different just scales. Click here for just tuning PDF.

* note: Pythagorean and meantone temperaments are not quite even. If they are extended in both directions into the chromatic notes, then the circle is closed with a 'wolf' interval, i.e. G♯ is not the same pitch as A♭. However, considering diatonic scales only, we do not need to worry about this at all.