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Adjusting your tangents

Fine Tempering

by Bart Brashers, with help from Paul Morrissett

First published in Nyckel Notes #1, Oct 1995, revised and re-published in issue #18, Feb 2000.

Every tangent (the little post that sticks up from a key and touches the string) on a nyckelharpa must be adjusted — tuned — for a nyckelharpa to play in tune. The bottom of the tangent is a cylinder, which is inserted into a hole in the key and can be rotated from left to right, changing where it stops the string and thus the pitch of the note. All nyckelharpas need their tangents tuned from time to time — professionals do it every time they change their strings. You should adjust them at least once every year or so. Changes in humidity also tend to make a harpa need touching up. Anders Mattsson and Puma were both observed to grab a tangent and turn it a bit until an interval sounded right.

You can tell when your tangents need attention by listening to your double-stops. Play some octaves (every other finger on the 1st and 2nd string) or C on the 2nd string with E on the 1st string, or G on the 2nd string with B and then D on the 1st string. Listen, and decide if they sound right. You can also tell if you play unison with someone else, and some of your notes’ pitches don’t match (assuming your open strings match!).

Adjusting your tangents

To adjust your harpa, first of all make sure that your bridge is in the right place. The distance between the inside edges of the nut and the bridge should be 40.0 cm (400 mm). Decide which fine-tuning scheme (see below) you want to use — for this example, I’ll use the Olsson fine-tuning scheme. Using an accurate tuning machine, tune your open A (1st) string 2 cents low. Using a pair of pliers to gently twist the 12th tangent (high A), tune it 2 cents low as well. It’s best if you try to grab the tangent as far down as possible — not right at the top — to minimize the twisting strain. Put the flat of your pliers against the flat side of the tangent that is opposite from the direction the tangent must go, squeeze gently, and turn. I use a pair of pliers that lack any teeth in its jaws, to avoid marking the tangents. I’ve seen others use pliers where the jaws have been filed smooth, or wrap the jaws with black electrical tape to spare the tangent. If the note is low, turn the tangent toward the bridge — if it’s high, turn it away from the bridge.

Repeat this for the high D and highest G tangents (and maybe a few more) and take a look at them. Are they more or less centered (straight) or are they all turned to one side? If too many of your tangents have to be turned far from straight, it can be problematic. If the force of the string is not along the direction of the key (making the tangent want to rotate in its hole, rather than simply push back), the note will not sound clean and strong. Try adjusting the placement of the bridge, and re-do your sample tangents. If the tangents angle toward the bridge, you should move the bridge closer to the keys (shorten the vibrating part of the strings). It’s okay to have the A-string be a different length than the G-string (old nyckelharpas all were that way). Just remember to measure and write down your new string length on a scrap of paper, and put it in your case, for the next time you adjust your harpa. On my harpa, the 1st string is 404 mm long, while the 4th string is 400 mm.

Now that you’ve got the bridge in the right place, tune all the tangents on the A string the appropriate amount high or low, following the table. Check the open string often, to make sure it doesn’t drift. Play each note only with a down-bow, since most people’s up-bow is slightly sharper than their down-bow. Then tune the open C string 4 cents high, and adjust those tangents. Do the same for the G string.

Since wood swells in humid air, it’s probably not a good idea to try to tune your tangents in the midwest summer (unless you keep your nyckelharpa in an air-conditioned room for a week or so first). There is a risk of splitting the tangent — if you twist with the pliers near the top of the tangent and the tangent is too swollen to twist in its hole, you could ruin the tangent. If that happens, it’s likely you’ll have to disassemble the whole keybox to remove and replace the tangent. It’s best to keep a few spare tangents in your nyckelharpa case. If you don’t have any, Bart Brashers has offered to send you some!

Daily tuning

Each time you pick up your Nyckelharpa to play, tune your A string 2 cents low, your C string 4 cents high and your G string 2 cents high. Check your tuning by playing several double stops — intervals you know you want to sound good. Find the best compromise that makes the intervals sound the best overall. Then, assuming you have time and it’s quiet enough, tune your resonance strings to your keys rather than to your tuner.

Fine-tuning (Tempering) Schemes

Just (Ptolemaic) Intonation
Tonic 2nd 3rd 4th 5th 6th 7th
1/1 9/8 5/4 4/3 3/2 5/3 15/8

This scheme is based on ratios of the notes of a diatonic scale to the tonic note. All three major chords, C, F, G, are purely resonant (within ± ½ cent) All the major triads are in ratios of 4:5:6 (e.g. C-E-G or F-A-C) or in the ratio 3:4:5 if the major 5th is put on the bottom (e.g. G-E-C or D-G-B). This is considered ‘perfect’ tuning, since all the chords and intervals sound really good, perfect in fact, to the human ear. There are no ‘beats’ within the tone, and the melodies sound glorious. The problem is, it only works for one key at a time. You could tune your nyckelharpa using this scheme, but as soon as you switched keys your intervals and your scale would be off.

Equal-Tempered Intonation
Interval* Compared with Just Intonation
5ths 2 cents too low
Major 3rds 14 cents too high
Minor 3rds 16 cents too low

*All intervals are the same — equal.

Electronic tuners use the equal-interval scale, which was developed so instruments can play in any key and still sound reasonably good. The “distance” between each of the notes is the same, regardless of where you are relative to the current tonic note (the current key). This makes the piano sound equally good in all keys, something that was important to people who want to change keys a bunch of times in the same piece. However, this also makes the piano sound equally bad in all keys. Fretted instruments like guitars also use this tempering. If you play a major third on a guitar or a piano, you can hear that the interval is a little high (sharp) — 14 cents high, as compared to the Just Intonation. That’s why it’s so hard to tune the 2nd string of a banjo. You can’t simultaneously make both the 3rd (open string) and the 5th (3rd fret) sound good against the G (open 3rd string).

Violinists can (and do) adjust their fingers as they play to temper their scales. Many old nyckelharpas in museums seem to have non-piano temperings, but we can’t be sure they haven’t been randomly adjusted by the passage of time and all the knocks the harpas have taken.

The thing is, nyckelharpa players don’t play in every key! We play in a select few keys: C, F, G, D, etc. and don’t play in the keys of G# or C# (at least most of us don’t, voluntarily). We are then free to optimize the tempering for the keys we commonly use. Since we don’t play any tunes is the key of F#, it doesn’t make sense for us to use a tempering that prioritizes that key as highly as F, C and G.

The Olsson fine-tuning scheme (Vallotti Intonation)

The Olsson fine-tuning scheme (Vallotti Intonation)
Click to Enlarge

John Olsson, a cantor and folk musician (and amateur folk researcher) from Björklinge chose a method of fine-tuning the tangents on the nyckelharpa that sounds best in the keys of C, G and F, the most common keys of the traditional music of Uppland. It turns out that this method has been known for a long time, and used to intonate most church organs in the 1700s (the music played on the organ has some similarities to some of the nyckelharpa’s repertoire). It is also the way many piano tuners tuned by ear, before the common use of tuning meters. 5ths are 2 cents off, but that makes the 3rds and 7ths a lot closer to the Just Intonation. It’s a compromise.

In my previous article on this subject (Nyckel Notes #1, Oct 1995) I had incorrectly attributed this scheme to Eric Sahlström, and called it “Eric’s fine-tuning (intonation) scheme”. Olov Johansson tells me that Eric was not so structured in his approach to fine-tuning, and treated each nyckelharpa a little differently. He arrived a something similar to the Olsson scheme, but Olov isn’t even sure Eric ever had a tuning machine.

The top line of the table contains the names of each note of the chromatic scale, in circle-of-fifths order. The second row contains the amount (in cents) to de-tune each note from what the tuner says is correct. 100 cents is one half-step, so you can see that the corrections are pretty small. The third row is the key priority rank – which key (not which note) will sound the most in-tune with itself.

For example, in the Olsson scheme, the key of C will sound the most in-tune with itself, and the key of F# will sound the least in-tune with itself (fortunately, we never play in the key of F#!). The note named A should be tuned 2 cents flat compared with the tuner, the note named C should be tuned 4 cents sharp, etc.

Bart’s Fine-tuning Scheme (a variation of Olsson method)

Bart's Fine-tuning Scheme
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I’ve been playing more and more tunes in A and D lately — playing at 3rd and 4th priority in the Olsson scheme. The slightly sour intervals have been bugging me, so I worked out a slight variation of the Olsson scheme. I simply bumped the `cent’ row one step to the right. This tuning will sound the best in the key of G, second best in D and C, third best in A and F, etc., (the key of C# will sound worst). Essentially, it prioritizes the key of D and A higher than F and Bb compared to the Olsson scheme. This scheme will make the “fiddle keys” of D and A sound better than the Olsson scheme. Plus (an added bonus) it’s easier to tune to the fiddle’s A string — or rather, to give an A to the fiddle player and make him/her re-tune, since we have more strings!

J.S. Bach’s Fine-tuning Scheme (Meantone Temperament)

J.S. Bach's Fine-tuning Scheme
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This is probably very close to the tuning used by J.S. Bach and people living around his time; it’s what they meant by “well tempered” in titles like “The Well Tempered Clavier”. Somewhat accidentally, it gets major 3rds just exactly right (5/4 resonance with the tonic) all the way up the diatonic scale, and all of the other chord intervals in the diatonic scale are at most off by 5.4 cents – some a little flat, some a little sharp. The drawback is that the accidentals (sharps and flats) are off by as much as 21.5 cents! Too much to sound even close to good.

Sören Åhker’s fine-tuning scheme

Sören Åhker's fine-tuning scheme
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That’s right, Sören’s method detunes (compared with the tuner) the tangents and the resonance strings by different amounts! The resonance strings are always a little lower (flatter) than the tangents. Olov Johansson (of Väsen fame) also tunes his understrings slightly lower than his tangents, so this is not completely unheard of. The idea here is that you don’t want too much response from your understrings, lest they overwhelm your melody and make the harpa sound ‘muddy’. Olov also talked about not having an instantaneous response from the resonance strings, preferring longer-term overtones and ringing.

The other important difference here is that this is the only scheme that is not based on mathematics, but on the human ear and how a nyckelharpa sounds. Sören developed this tuning scheme by adjusting and listening until it sounded good, and then measured it and converted it to cents. He repeated the exercise with Peter Puma Hedlund, measuring his harpa after he intonated it by ear, and got largely the same results. This scheme makes the nyckelharpa ring the best, he says.

Chords and intervals

Paul Morrissett did the math of the chords (double-stops) in a few of the above intonation schemes. The lower note was always taken to be the tonic — 0 cent wrong by definition. The intervals tonic to 5ths (written 5th), tonic to major 3rds (written +3rd), tonic to minor 3rds (written -3rd), major 3rds to 5ths, and minor 3rds to 5ths are then ‘wrong’ compared with the Just Intonation by the following amount:

the math of chords
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For C, F and G chords in the Olsson scheme, having the 5ths 2 cents flatter than with equal temperment allows the thirds to be 8 cents flatter, which is substantially closer to Just Intonation. Sören’s method looks pretty good for the key of C, but inferior to the Olsson method for the keys of Bb, F, G, and D, and much worse in Eb and C minor (important for those of us who play Ödetorpsvalsen).

Perhaps another way to evaluate this table is to count the number of red and/or yellow boxes, and compare it with the number of green and/or blue boxes. Or perhaps compare the Total Error, the sum of the absolute values of the cents wrong from Just Intonation in all the non-gray boxes?

Total error comparison
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First of all, you can see that either of these beats just using the Equal-Tempered intonation, since Green + Blue (better) outweighs Red + Yellow (worse). Sören’s method has more intervals where it does no better than the Equal Temperament (White). In fact, it does worse (Red + Yellow) in almost as many cases as it does better (Green + Blue). Despite the fact that the Olsson method does worse in more cases than Sören’s method, it does better in significantly more cases than Sören’s method. The total error is also lower for the Olsson scheme, though not by all that much (Sören’s scheme is only 13% more).

Many thanks to Paul for doing the math and sharing his results! You can also use this java-based tool to calculate offsets compared to the Just Intonation (and listen to various intonations).

Combined table

Finally, I’ve found the tables above a bit hard to use, since it’s hard to look up a particular note in circle-of-fifths order to find its offset. So here’s a re-sorted and combined version of the tables above — I printed this article, cut this table out, and keep the slip of paper in my nyckelharpa case.

Combined Table
Click to Enlarge

More information on temperments:

Understanding Temperaments
Tom Lougheed’s analysis of temperments
Terry Blackburn’s guide to alternate temperments