Temperament

Every once in a while someone will write us asking about information related to early music. We recently received an e-mail asking for more information on our meantone tempered guitar. Thus, after many years, it is time to post some information.

Let us start from the very beginning. What is temperament? It is a system of mistuning. This is not very informative and takes some explanation of what it means to be in tune, and why it is necessary to mistune some notes.

We all have some experience of the concept of music being ‘out-of-tune’. When two tones are not in tune we hear what is known as a beat, a repeating throbbing sound. If the note is nearly in tune, this beat can be very slow, such as less than once per second. If it is very out of tune, the beat produces a warbling effect as it recurs very frequently.

To gain more insight, let us examine from a mathematical perspective what it means for notes to be in tune, and by extension what is means for them to be out of tune. Musicians have considered it from this perspective for thousands of years, and is closely associated with the work of the ancient Greek mathematician Pythagoras. In the Medieval era the tuning of tones was demonstrated with a device known as a monochord, which is so named on account of its single string. This lays over two end points and is put under tension. A bridge is placed underneath the string so that it is divided into two sections. Each of these sections produce a pitch when plucked. By adjusting the placement of the bridge the pitches produced by the sections of the string change. When the two sounds are in tune, the ratio of the lengths of the string sections when reduced to lowest terms consists of small integers. Such an interval would be called ‘pure’, or being in ‘just intonation’.

The ratios identified as being pure or in tune are as follows:

Unison is 1:1

Octave is 2:1

Perfect 5^th is 3:2

Perfect 4^th is 4:3

Major 3sup>rd is 5:4

etc.

Up to this point, the musical mathematical world seems very much in order. However, everything in this mathematical system does not work out so nicely. Many of these ratios when taken in sequence are not commensurate with one another.

Let us consider the fifths that can occur in a given octave.

C g

g d

d a

a e

e b

b f#

f# c#

c# g#

g# d#

d# a#

a# e#(f)

e# b# (c)

In a perfect world, if these intervals were played as an ascending sequence, the ratio of the lowest c to the highest c would be equal to 2:1 to the 7^th, as the high c is seven octaves above the low one, and the ratio for a pure octave is 2:1. This ratio is 128:1

If these intervals are tuned as a sequence of 12 ascending pure fifths we have 3/2 to the 12^th. This results in the following ratio between the bottom c and the high c: 531441:4096.

For the sake of comparison 128:1 can be expressed as 524288:4096. This is obviously smaller than 531441:4096. Thus 3/2 to the 12^th does not equal 2/1 to the 7^th. The sequence of pure fifths is slightly larger than seven octaves.

Their ratio is equal to 531441:524288 and is known as the Pythagorean comma. (The term comma is derived from the Greek term κόμμα. This in turn comes from the term κόπτω meaning to strike, beat, or cut, and the suffix μᾰ, turning it into a noun. Other meanings of the term refer to short clauses, and even decimal places.)

When expressed as a decimal ratio it is 1.0136432647705078125. (A unison is equal to 1.)

(It should be noted that although the argument presented above uses perfect fifths, ancient texts used arguments based on whole tones 9:8. Six whole tones are larger than an octave by the amount of a Pythagorean comma.)

The Pythagorean comma was not the only comma related to music. Another is the syntonic comma, which is the difference between a pure major third, and four pure fifths minus two octaves. This has a ratio of 81/80. In this we see it is not only the ratio of octaves and fifths that are not commensurate. In musical settings, if the term comma is used without qualification, it refers to the syntonic comma.

In Western music, it is paramount to have octaves that are pure, thus some of the fifths (and thirds) must be out of tune. ‘Mistuning’ these intervals is known as tempering, and a system of such mistuning is known as a temperament. And thus we arrive at an understanding of the statement from the beginning of this document.

The first known temperament is simply to have one of the fifths reduced by the comma. It is known as Pythagorean tuning/or Pythagorean temperament. The out of tune fifth was known as the wolf, because it howled.

From the perspective of tuning an instrument, Pythagorean temperament can be viewed as a process of tuning pure fifths. You can begin on a note, and tune the note that is a fifth above it. From a musical perspective it would not be desirable to tune 12 ascending perfect fifths. This would result in the fifth below the tonic to be mistuned. Thus, in practice the temperament is set by tuning five ascending fifths from the tonic, and six descending fifths from the tonic. Thus, the tonic note, and the sequence of tuning is important to the resulting sound of the music. This disposition of Pythagorean temperament was given by music theorist Arnout von Zwolle who lived in the mid part of the fifteenth century.

Over the course of history musicians explored many different temperaments. One class of temperaments that were popular in the Renaissance were meantone temperaments. Such temperaments had fifths that were smaller than pure fifths. In ¼ comma meantone each fifth was ¼ syntonic comma too small. This resulted in a sequence of four fifths such that the resulting third is pure. For example: F, c, g, d, a. The third F to a was pure.

In practice when setting the temperament (tuning an instrument) by ear, a novice may tune a pure fifth and then decrease its size slightly. The ‘pulse’ of the beat is noted, and compared to the pulse of the beat of the next fifth. One wants them to be identical. The sequence of four fifths is checked to ensure that it results in a pure major third. If it is not, then the tuning of the fifths must be revised. Once F, c, g, d, and a are tuned, the temperament is completed by tuning several pure thirds: c-e, d-f#, d-bb, e-g#, g-eb, a-c#, and g-b.

There is of course much more to say. I want to cut to the chase, and address creating the meantone guitar. One cannot simply tune a guitar in meantone temperament, as the pitches produced by the instrument are also governed by the placement of the frets. Thus, if one wanted to convert an existing guitar to meantone temperament ones would need to adjust the placement of the frets. This is most easily accomplished by creating a new fretboard and placing the frets appropriately. (One could also remove the frets, fill in the grooves with sawdust and glue and then place new frets.)

The lengths of the strings for the temperament can be determined by the ratios given in the temperament. Thus we would have to measure the length of each string and divide it per the ratios. (Note that the strings fan out, and that the bass and treble strings are typically slightly longer than the middle strings. Also note that it is possible to run into trouble by trusting numbers if you do not know what they mean. For example a guitar may have a scale length of 650mm. However, this does not mean that the length of the strings is 650mm. Sometimes the saddle is set back a small amount to correct the intonation of the guitar and thus the length of the string is actually longer than the listed scale length. Further, sometimes the numbers provided by string manufacturers are not accurate. My advice is to measure everything several times.) However, the story does not end there and things get much more complicated. The pitch produced by a string is influenced by other factors than its length. Notable among these is its tension. When a guitar string is pressed down on a fret, its tension is increased slightly. The effect of this increase is related to the height of the action on the specific fret of the guitar (the distance that the string is depressed), the length of the vibrating string section, and properties of the string, such as its thickness, mass, elasticity / Young’s modulus. It is possible to obtain a more accurate estimate of the fret placement by taking these things into account. However it is a whole lot of work and is still not wholly accurate. In practice it is easier to empirically set the frets starting from the placement based on the ratios. Those placements will be a bit sharp, as when the string is depressed it will sound a bit higher. Thus it is necessary to move the fret back slightly. To determine how much to move it, pluck the string and measure the frequency that it produces. If it is too high, then move the fret towards the nut. If it is too low, move the fret towards the saddle. Once the proper placement is determined, one can then route out the groves to attach the frets. (It is not recommended to saw them (as is traditionally done), as the frets will not necessarily run across the entire fretboard.)

I can’t resist discussing the details of the calculation. (However, please take my advice and place the frets empirically.)

You will need to obtain the Young’s modulus for the specific strings that you use (the manufacturers will not provide them as they consider them to be trade secrets), measure their mass per unit length, and measure their core/cross sectional area.

Young’s modulus is the tensile stress divided by the longitudinal strain. The formula for the tensile stress is force divided by area. For the area we will approximate the actual area with that of a cylinder. (The actual string may consist of several cords that are wound together) Thus we measure its diameter, and use that to obtain the radius and use the π r squared to obtain the area at a given point. This is then multiplied by the length of the string. The formula for the longitudinal strain is (length under strain – length) / length.

Tension stressed is given by the following equation: T stressed = ((L when fretted – L unstressed (at that fret) ) / L unstressed (at that fret) ) * E * A

E=Young's modulus

A=core area of string

That is the increase in the amount of tension. You will need to add the existing tension.

Then you can put it into an equation to calculate the frequency.

f = (1/2L)*√(T/μ)

f is the frequency in hertz (Hz) or cycles per second

T is the tension

L is the length of the string section

μ is the mass per unit length of the string section

For those of you who love this sort of thing, there are many resources to help with calculations, although again I will say that they will not produce a better result than the empirical one of testing the frequency. One resource is the book Contemporary Acoustic Guitar Design and Build by Trevor Gore and Gerard Gilet. Another is the paper “A Pitch Error Model for Fretted String Instruments” by John Lane and Takis Kasparis. It presents a model for pitch error based on the increase in tension of fretting a string.

OK, that is enough of that. Here is some practical data to get you started calculated to many too many decimal places.

¼ comma meantone temperament information

A=440

tune C# and F in pure thirds from A

tune D, G, and C in tempered fourths/fifths from A

tune pure thirds from those notes.

tune G# from E derived from C. (rather than down from C)

Crucial values

M3 up 1.25

M3 down 0.8

m6 up 0.625

m6 down 1.6

Tempered 5th up

1.4953487812212205419118989941409133953634597576147

Tempered 5th down

0.66874030497642202400323307325864793638803519165248

¼ Comma Meantone Frequencies calculated as described above

Low E string and high E string

650 mm string length

Fret position from nut in mm

A string (11 fret is different from E)

D string chart (6, 11, and 18th frets are different from E)

G string chart (1, 6, 13, and 18 frets are different from E)

B string (4, and 16^th frets are different from E)