pythagorean temperament for lute

7/29/2019 Pythagorean Temperament for Lute

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Pythagorean temperament for lute

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

In Pythagorean temperament, fifths are pure. A consequence of pure fifths in atemperamant for fretted instruments is that major thirds are much too wide, andminor thirds much too narrow. That makes Pythagorean temperament suitable formusic in which the fifth is an important consonant and in which the third is not muchused in triads. This is true of part of the medieval repertoire, hence many musiciansprefer to play this music in Pythagorean temperament.

In this article I will explain about Pythagorean temperament and show you how toreposition your frets and tune your courses. Finally, I will explain how to put your

frets back into their positions for equal temperament.

Pythagorean temperamentRepositioning of fretsTuningBack 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.

Pythagorean temperament

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)

In equal temperament a fifth is 700 cents, a major third 400 cents and a minor third300 cents. These are compromises most people can adjust their ears to. InPyhtagorean temperament the fifth is pure, hence 701,96 cents. A major third is407,84 cents and a minor third 294,12 cents, both even further away from purethan in equal temperament.


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Here is a circle of fifths, with cents deviations from equal temperament written nextto each note.

Notice that E-flat is not the same as D-sharp, as it is in equal temperament. You canalso not substitute G-sharp with A-flat, etc. And although all other fifths are pure,the fifth between G-sharp and E-flat, or D-sharp and B-flat, depending on whichpitch you have chosen to tune the note, is very un-pure: only 678.44 cents. You

cannot adjust your ears to this compromise anymore, this fifth is simply too smalland very much out of tune. It is called the wolf.

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Repositioning of frets

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 way

you 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 Pythagorean temperament, we can calculate fret positions. Here is atable with the deviations from equal temperament of each fret, incorporating thedeviations from equal temperament of the open courses:

open1st fret

2nd 3rd 4th 5th 6th 7th8th fret


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string fret fret fret fret fret fret

G + 3.9G# +13.7

A +3.9

Bb-5.9

B +7.8

C -2.0

C# +11.8

D +2.0

Eb -7.8

D - 2.0Eb -

9.8

E +

3.9F - 5.9

F#

+7.8

G -

2.0

G# +

11.8

A +

2.0

Bb -

7.8

A + 0Bb -9.8

B +3.9

C - 5.9C# +7.8

D -2.0

Eb -11.8

E +2.0

F - 7.8

F - 7.8F# +13.7

G +3.9

G# +17.6

A +7.8

Bb -2.0

B +11.8

C +2.0

C# +15.7

C - 5.9C# +13.7

D +3.9

Eb -5.9

E +7.8

F -2.0

F# +11.8

G +2.0

G# +15.7

G + 3.9G# +13.7

A +3.9

Bb-5.9

B +7.8

C -2.0

C# +11.8

D +2.0

Eb -7.8

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

0.0510.064

0.111 0.156 0.210 0.2500.2880.298

0.3330.3670.376

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

3.03.6

14.7 9.2 12.4 14.817.017.6

19.721.722.2

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


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For fret b (first), g (sixth) and i(eighth) we come up with two fret positions: one fornotes that are lower than in equal temperament and one for notes that are higherthan in equal temperament. On the first fret the lower notes are the e-flat and b-flaton the second and third courses. The higher notes on the first fret are the g-sharpon courses one and six, the f-sharp on the fourth course and the c-sharp on thefifth course. If you want to use notes from both these two groups in one piece, youhave to find a solution. One solution is to put the fret in the high position and to glue

a little extra fret (called tastini, Italian for little fret) in the lower position for only thesecond and third courses. You can use a small bit of fret or matchstick and sometape. A friend of mine uses little custom made ebony wedges. Another solutionmight be to use two frets.The g- and i-frets (six and eighth) cause usually no problems, you can place themin either the low or the high position, depending on the piece you play. Just try andlisten.

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Tuning

Now you only have the fret positions for Pythagorean temperament. Next you willhave to tune the open courses of the lute, too. Notice that you do not need a tunerat all, nor, actually, the explanation above to move your frets, as tuning in pure fifthscan safely be done by ear alone:* 1. Tune the third course, a, on a tuning fork.* 2. Tune the second course, d, by tuning a pure fourth with the third course. You

can use the harmonic above the fifth fret of the third course and the seventh fret ofthe second course.* 3. Tune the first course, g, by tuning a pure fourth with the second course. Youcan use the harmonic above the fifth fret of the second course and the seventh fretof the first course.* 4. Tune the sixth course, g, by tuning a pure double octave with the first course.* 5. Check the position of the second fret by comparing a (second fret) on the firstcourse with the open third course, and the on the second course (second fret) withthe pure fifth on a (or double octave with the harmonic on the seventh fret of thethird course).

* 6. Tune the fifth course, c, by tuning a pure fourth with the open sixth course. Youcan use harmonics again.* 7. Tune the fourth course, f, by tuning a pure fourth with the open sixth course.You can use harmonics again. Check the octave on the second fret, g, with theopen first course.* 8. Check third fret with f and c on second and third courses with open fourth andfifth courses.* 9. Check fourth fret by comparing a on fourth course (fourth fret) with the openthird course.


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* 10. Adjust first fret to either flats or sharps (or add tastini). Flats: octaves e-flatand b-flat on second and third courses with third fret on fifth and first/sixth courses.Sharps: octave f-sharp on second course (fourth fret) with first fret on fourthcourse.* 11. Check pure fiths and octaves all over the fingerboard in the positions and keysthat you will use. Adjust here neccecary and avoid where no soution can be found(e.g. g-sharp on fourth course, third fret).

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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.

More on temperaments for lute in Mean-tone temperament for lute.

David van Ooijen 6/[email protected]

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pythagorean temperament for lute

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