Mean-tone temperament for lute

The examples in this article are for a renaissance lute in g'.

All through music history several systems of tuning were used. Some of thesetunings, also known as temperaments, sound very well on renaissance lutes.Mean-tone temperament, in which you have pure thirds, is especially good. It is notso difficult to tune your lute in mean-tone temperament, and the result is absolutelyworthwhile. In this article I will explain about mean-tone temperament and show youhow to reposition your frets and tune your strings for it. Finally, I will explain how toput your frets back into their positions for equal temperament.

Pure tuning is impossibleEqual temperamentMean-tone temperamentRepositioning of fretsTuningOne sixth comma mean-tone temperamentSilbermannBack to equal temperament

In this article I will use cents. One cent is defined as the 100th part of the differencein pitch between two notes half a tone apart in equal temperament. It is easier touse cents then frequencies in Hertz when writing about temperaments.

Pure tuning is impossible

As Pythagoras already knew in the 5th century B.C., pure intervals are naturalharmonics and therefore simple relations between frequency ratios:

1 : 2 octave (1200 cents)2 : 3 fifth (701.96 cents)3 : 4 fourth (498.04 cents)4 : 5 major third (386.31 cents)5 : 6 minor third (315.64 cents)

If you take a c, and add 12 pure fifths to it, you will get a b-sharp6. The frequencyratio between c and b-sharp6 is 12 times 2 : 3 equals 4096 : 531441.If you take the same c, and add 7 pure octaves, you will get a c7. The frequencyratio between c and c7 is 7 times 1 : 2 equals 4096 : 524288.The pitches b-sharp6 and c7 are enharmonically the same; they share the same

key on the piano and the same fret on the lute. However, the result of calculatingthem via pure fifths or via pure octaves is not the same. The difference is called thePythagorean comma. It is exactly 23.46 cents.To make a workable system of tuning for the lute, after all, you can have only onefret for a b-sharp and a c, this Pythagorean comma has to be gotten rid of. To thinkabout this, it helps to imagine a circle of fifths:

Back to top

Equal temperament

The easiest way of spreading out the Pythagorean comma is to divide it equallyover all 12 fifths. This system of tuning is called equal temperament and moderninstruments like guitars and pianos are tuned in equal temperament. In equaltemperament all fifths are 23.46 divided by 12, equals about 2 cents, too small (700cents in stead of 702 cents). This is acceptable for our ears. The major thirds,however, are much too big: 400 cents instead of 386 cents. This is not verybeautiful. If you play a harmonic around the fourth fret, you will hear a pure majorthird, try it on the fourth string, then you will hear an a. Compare this to the equalthird, the normal a on the second fret on the first string, and you will hear thedifference. Historical evidence suggests equal temperament was used for lutes fromabout 1550 onward, though Vincenzo Galilei still fulminates against the use of socalled ‘tastini’ (little frets) in his Fronimo printed in 1568 and reprinted in 1584. Heexplains that lute players of his time add these to ‘remove some of the sharpnessfrom the thirds and major tenths’. To me, that reads like an advertisement in favourof mean-tone temperament; who wouldn’t want the sharpness from his thirdsremoved? Early 17th century Italian music with wind instruments sounds best inmean-tone temperament. Many organs are still tuned in mean-tone temperament upto the present day.

Back to top

Mean-tone temperament

If we add up five pure fifths, e.g. c - e2, the ratio between the c and the e2 is 16 :81, the third c2 - e2 is 64 : 81. A pure major third, however, is 4 : 5. The differencebetween these two thirds is 80 : 81, or 21.50 cents. It is called the syntonic comma.To make a temperament in which major thirds are pure, this syntonic comma has tobe gotten rid of.

Mean-tone temperament does just that: it makes 11 fifths each one quarter syntoniccomma too small. The remaining fifth will be much too big. The 11 fifths are audiblytoo small, but still usable. The 12th fifth, however, is unusable and is called the wolf.It is sometimes used to express very harsh words in the text. Usually the wolf isbetween the g-sharp and the e-flat, or between the d-sharp and the b-flat,depending on which note is needed in the music: the d-sharp or the e-flat. Inmean-tone temperament, all thirds are pure, which gives a wonderfully warm,sonorous sound when playing chords. A lute tuned in mean-tone temperamentsounds a lot better. Other systems of mean-tone temperament do not tune the 11fifths a quarter comma too small, but for example one sixth comma (Silbermann) ortwo seventh (Zarlino 1558). If you have a programmable tuning machine (like theKorg MT-1200 or the Sonic Research ST-122) you can enter the following centsdeviations from equal temperament for each note:

Back to top

Repositioning of frets

But you don’t need a tuning machine to tune your lute in mean-tone temperament.

First, take a piece of paper and place it on top of the fingerboard of the lute, oneend touching the nut. Now mark the current fret positions on the paper. This wayyou can put your frets back in their original position later.Now that we know how many cents higher or lower than equal temperament a notehas to be in mean-tone temperament, we can calculate fret positions. Becausemean-tone temperament is a temperament with equal fifths, the fret positions ondifferent strings are the same. Here is a table with the deviations from equaltemperament of each fret, incorporating the deviations from equal temperament ofthe open strings:

openstring

1st fret2ndfret

3rdfret

4thfret

5thfret

6thfret

7thfret

8thfret

G + 6.8G# -23.9

A -6.8

Bb +10.3

B -13.6

C +3.5

C# -20.5

D -3.4

Eb +13.7

D + 3.4Eb +17.1

E -6.8

F +10.3

F# -13.7

G +3.4

G# -20.5

A -3.4

Bb +13.7

A + 0Bb +17.1

B -6.8

C +10.3

C# -13.7

D +3.4

Eb +20.5

E -3.4

F +13.7

F +13.7

F# -24

G -6.9

G# -30.8

A -13.7

Bb +3.4

B -20.5

C -3.4

C# -27.4

C +10.3

C# -24

D -6.9

Eb +10.2

E -13.7

F +3.4

F# -20.6

G -3.5

G# -27.4

G + 6.8G# -23.9

A -6.8

Bb +10.3

B -13.6

C +3.5

C# -20.5

D -3.4

Eb +13.7

This results in the following list of numbers, with which you will have to multiply thestring length of your lute to get the new fret positions:

b c d e f g h i

0.0430.065

0.106 0.164 0.200 0.252 0.284 0.3310.3600.375

For a lute with a string length of 59 cm this would result in the following fret positions

in cm:

b c d e f g h i

2.53.9

6.2 9.7 11.8 14.9 16.8 19.621.122.1

You can mark these fret positions on a similar piece of paper again.

For fret b (first) and i (eighth) we come up with two fret positions: one for notes thatare higher than in equal temperament and one for notes that are lower than inequal temperament. On the first fret the higher notes are the b-flat and e-flat on thesecond and third string. The lower notes on the first fret are the g-sharp on stringone and six, the f-sharp on the fourth string and the c-sharp on the fifth string.Some 16th century lute music takes this into account: it will use either notes for ahigh first fret, or for a low first fret. Notes from both groups will not occur in onepiece, so you can reposition the first fret according to the piece you are playing. Butif you want to use notes from both these two groups in one piece, you have to find asolution. One solution is to put the first fret obliquely: low for the lower strings, highfor the higher strings. This way we can play the notes on string five and six more orless in tune, and hopefully those on string two and three as well. We cannot use thefirst fret on the first string, however, so this is not an ideal solution. A better plan isto put the fret in the low position and to glue a little extra fret (called tastini, Italianfor ‘little frets’) in the higher position for only the second and third string. You canuse a small bit of fret or matchstick and some tape. A historic solution might be touse a double first fret, like on viola da gambas, and split the fret to cover bothpositions. I have not tried this. Another solution is to make two frets, the lower oneof thin carbon or nylon with little bits of electricity insulation cable around it. The littlebits of insulation cable can be placed under the strings that need it. It takes someexperimenting with thickness of material to get tastini that do not buzz, but this wayyou can still adjust the position of the tastini, choose which strings you want themunder, and finally, push the whole fret against its neighbour when you do not wantto play in mean-tone temperament any more. A friend of mine uses little custommade ebony wedges.The i-fret (eighth) is usually no problem, you can place it in either the low or thehigh position, depending on the piece you play. Just try and listen.

Back to top

Tuning

Now you only have the fret positions for mean-tone temperament. Next you will have

to tune the open strings of the lute, too. This is simple:* 1. Tune the third string, a, on a tuning fork or tuning machine.* 2. Tune a pure major third downward, a-f, between the third and fourth strings.You can use the new fourth fret position, or use the harmonic above the fourth fretof the fourth string to get a pure third.* 3. Tune the second string, d, by tuning its third fret, f, in a pure octave with theopen fourth string.* 4. Tune the fifth string, c, by tuning the open string in a pure octave with the thirdfret on the third string.* 5. Tune the first string, g, by tuning the second fret, a, in a pure octave with theopen third string.* 6. Tune the sixth string, g, by tuning a pure double octave with the first string.

You are now ready to play in mean-tone temperament. You will notice that mostchords sound very warm. Maybe some frets will need a little adjustment. Somenotes are not usable, try the third fret on the fourth string in an E-major chord(example 1): horrible!

Or try E-major with a g-sharp on the first fret of the first string (example 2). Now tryf-minor with an a-flat on the first fret of the first string (example 3).

Depending on you having made a low first fret (g-sharp) or a high first fret (a-flat),one chord will sound good, the other bad.

Back to top

One sixth comma mean-tone temperament

Because many lute players prefer one sixth comma mean-tone for 16th and early17th century repertoire, as it is not so extreme as one quarter comma or ‘true’mean-tone temperament, I will give cents deviations from equal temperament foreach note for the programmable tuning machines.This temperament is often called Silbermann temperament, but it is not, as

Silbermann used the pythagorean comma for his calculations, resulting in a fifth of698.045 cents, as opposed to the the fifth of 698,375 cents in one sixth syntoniccomma mean-tone temperament as described here.

Here is the table with fret displacement compared to equal temperament, expressedin cents:

openstring

1st fret2ndfret

3rd fret4thfret

5thfret

6thfret

7thfret

8thfret

G + 3.3G# -11.4

A -3.3

Bb +4.8

B -6.6

C +1.6

C# -9.8

D -1.7

Eb +6.5

D + 1.6Eb +8.2

E -3.2

F + 4.9F# -6.5

G +1.7

G# -9.7

A -1.6

Bb +6.5

A + 0Bb +8.1

B -3.3

C + 4.9C# -6.5

D +1.6

Eb +9.8

E -1.6

F +6.5

F + 6.5F# -11.4

G -3.2

G# -14.6

A -6.5

Bb +1.6

B -9.8

C -1.6

C# -13

C + 4.9C# -11.4

D -3.3

Eb +4.9

E -6.5

F +1.6

F# -9.8

G -1.6

G# -13

G + 3.3G# -11.4

A -3.3

Bb +4.8

B -6.6

C +1.6

C# -9.8

D -1.7

Eb +6.5

And here are the numbers by which you have to multiply your string length to getfret positions for one sixth syntonic comma mean-tone temperament:

b c d e f g h i

0.0500.061

0.107 0.161 0.203 0.252 0.289 0.3320.3660.372

For a lute with a string length of 59 cm this would result in the following fret positionsin cm:

b c d e f g h i

2.93.6

6.3 9.5 12.0 14.8 17.0 19.621.622.0

Here you can see the difference between the two frets in the first position, theb-fret, and the eighth position, the i-fret, are not so great any more.

And this is what it looks like on my 59cm 8-course lute. Notice the extra first fret forcourses four to seven, and the compromise tastini for g-sharp/a-flat in the thirdposition on the fourth course. I cannot play a g-sharp on the first fret of the firstcourse in this configuration, so I must be carefull with E-major chords.

Ross W. Duffin argues a strong case in favour of 1/6 syntonic comma mean-tonetemperament for baroque ensemble playing in his articles Why I hate Valotti (or wasit Young?) and Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma

mean-tone.

Back to top

Silbermann

Mainly because I have to play in Silbermann temperament myself quite often, andbecause there is some confusion with 1/6 comma mean-tone temperament, I haveincluded the calculations for Silbermann temperament. But the difference betweenthe two temperaments is very small.

Silbermann temperament, or Silbermann 1/6 comma temperament - there isconsiderable confusion over the names of the various temperaments associatedwith him - is one sixth pythagorean comma mean-tone temperament, as opposed tothe one sixth syntonic comma mean-tone temperament described above. ThePythagorean comma is 23.46 cents, a little more than the Syntonic comma of 21.50cents. One sixth Pythagorean comma is 3.91 cents. All pure fifths of 701.96 centsare tuned this 3.96 cents too small, resulting in fifths of 698.05 cents. That is but alittle smaller than the fifth of 698.38 cents in one sixth syntonic comma mean-tonetemperament described above. A fifth of 698.05 cents is 1.95 cents smaller than anequal temperament fifth of 700 cents.

Here is the table with fret displacement compared to equal temperament, expressedin cents:

openstring

1st fret2ndfret

3rdfret

4thfret

5thfret

6thfret

7thfret

8thfret

G + 3.9G# -13.7

A -3.9

Bb +5.9

B -7.8

C +2.0

C# -11.7

D -2.0

Eb +7.8

D + 2.0Eb +9.8

E -3.9

F +5.9

F# -7.8

G +2.0

G# -11.7

A -2.0

Bb +7.8

A + 0Bb +9.8

B -3.9

C +5.9

C# -7.8

D +2.0

Eb +11.7

E -2.0

F +7.8

F + 7.8F# -13.7

G -3.9

G# -17.6

A -7.8

Bb +2.0

B -11.7

C -2.0

C# -15.6

C + 5.9C# -13.7

D -3.9

Eb +5.9

E -7.8

F +2.0

F# -11.7

G -2.0

G# -15.7

G + 3.9G# -13.7

A -3.9

Bb +5.9

B -7.8

C +2.0

C# -11.7

D -2.0

Eb +7.8

And here are the numbers by which you have to multiply your string length to getfret positions for one Silbermann temperament:

b c d e f g h i

0.0490.061

0.107 0.162 0.203 0.252 0.289 0.3320.3640.373

For a lute with a string length of 59 cm this would result in the following fret positionsin cm:

b c d e f g h i

2.93.6

6.3 9.5 12.0 14.9 17.0 19.621.522.0

As you can see, in the final result for a 59cm lute, there is virtually no differencewith 1/6 syntonic comma mean-tone.

Back to top

Back to equal temperament

To put your frets back into their equal temperament positions, you can either usethe piece of paper on which you marked them, or calculate their exact positions bymultiplying the string length of your lute with the following numbers:

b c d e f g h i

0.056 0.109 0.159 0.206 0.251 0.293 0.333 0.370

Again, for a lute with a string length of 59 cm this will result in the following fretpositions in cm:

b c d e f g h i

3.3 6.4 9.4 12.2 14.8 17.3 19.6 21.8

Do not forget to tune the open strings back as well.

A good book for further reading is Mark Lindley’s Lutes, Viols and Temperaments(Cambridge University Press, 1984).

David van Ooijen 8/2001d.v.ooijen@planet.nl

This article first appeared in Nostalgia, the news letter of the Lute & Early GuitarSociety of Japan.

Back to Writings

Back to top