ELEMENTARY LUTHIERY

Adjusting the intonation is necessary to have the most in-tune instrument possible.

1. Tune the string to its pitch.
2. With a precise electronic tuner, check the pitch of the 12^th fret.
   • If it is lower than the pitch of the harmonic at the 12^th fret, move the saddle towards the neck;
   • If it is higher than the pitch of the harmonic at the 12^th fret, move the saddle away from the neck.
3. Repeat the operation for each string.

The action is the gap between the top of the 12^th fret and the bottom of the string. It is determinant in the playability of the instrument. It is set, once the trussrod has been adjusted, by raising or lowering the saddles. Recommended actions are given in Table 2. This table gives a range of values. The lowest values are achievable only on a recent, good quality instrument.

Table 2: Indicative action for guitars and basses, depending on the fingerboard radius.

	Radius 7.25 in	Radius 9.5 to 12 in	Radius 15 to 17 in
Guitar, low E	2 mm < 2.2 mm < 2.5 mm	1.6 mm < 2.2 mm < 2.5 mm	1.6 mm < 2.2 mm < 2.5 mm
Guitar, high E	1.6 mm < 1.75 mm < 2 mm	1.6 mm < 1.75 mm < 2 mm	1.6 mm < 1.75 mm < 2 mm
Bass, low E	2.8 mm	2.4 mm	2.4 mm
Bass, high G	2.4 mm	2 mm	2 mm

There is an approximate relation, for ideal strings, between the string tension, \(F\), the pitch frequency, \(\nu\), the length of the vibrating string, \(L\), and its diameter, \(d\):

\begin{equation} F = \pi \rho d^2 L^2 \nu^2, \end{equation}

where ρ is the density of the string material (\(\rho\simeq7850\) kg/m³ for steel). These tensions are usually expressed in kg at the sea level, \(F [kg]=F [N]/g\), with \(g=9.81\) m/s².

Eq. 1 applies to open strings as well as fretted strings. The vibrating length of an open string is called the scale length. It depends on the model of the guitar. Some manufacturers have different preferences. The two most common guitar scale lengths are:

Gibson:

L = 24.75 in = 628.7 mm;

Fender:

L = 25.5 in = 647.7 mm.

• Eq. 1 shows that string bending, which consists in increasing string tension by pushing it up or down, leads to \(F\propto{}d^{2}\nu^2\). It means that to bend a string by \(N\) half-steps, the string tension increases by a factor \(\delta{}F/F=2^{N/6}-1\), which is 12% for a half-step, 26% for a whole step and 41% for a minor third.
• It also shows that, to achieve bending to a given pitch on two different string gauges, the increase of tension is \(\delta{}F/F=(d_2/d_1)^2-1\), which is 56% going form \(d_1=0.008\) in to \(d_2=0.010\) in, and 125% going from \(d_1=0.008\) in to \(d_2=0.012\) in. In other words, bending a string requires to apply more than twice the strength with a gauge 0.012, compared to 0.008.

Eq. 1 shows that increasing the string gauge increases the tension by a factor \(\delta{}F/F=(d_2/d_1)^2-1\). Table 4 gives the tensions for a few typical electric guitar sets. It shows that a 0.012 string set applies twice more tension on the neck than a 0.008 set. It also means that playing 0.012 gauges requires twice more strength than 0.008 gauges.

Table 4: ideal string tension as a function of gauge, for typical string sets, for a Fender scale length (\(L=647.7\) mm), with steel strings.

String	Frequency	0.008 in string set	0.010 in string set	0.012 in string set
E (high)	330 Hz	F=4.74 kg (d=0.008 in)	F=7.41 kg (d=0.010 in)	F=10.7 kg (d=0.012 in)
B	247 Hz	F=4.15 kg (d=0.010 in)	F=7.02 kg (d=0.013 in)	F=10.6 kg (d=0.016 in)
G	196 Hz	F=5.88 kg (d=0.015 in)	F=7.55 kg (d=0.017 in)	F=13.8 kg (d=0.023 in)
D	147 Hz	F=6.48 kg (d=0.021 in)	F=9.94 kg (d=0.026 in)	F=14.1 kg (d=0.031 in)
A	110 Hz	F=7.41 kg (d=0.030 in)	F=10.7 kg (d=0.036 in)	F=13.8 kg (d=0.041 in)
E	82 Hz	F=6.61 kg (d=0.038 in)	F=9.68 kg (d=0.046 in)	F=11.9 kg (d=0.051 in)
Total		F=35.3 kg	F=52.3 kg	F=74.9 kg

It is interesting to know what is the effect of detuning on the string tension. In particular, it is useful to know the rough standard-tuning equivalent string gauge in an alternate tuning. For a given scale length, if we want to keep the string tension F, by decreasing its frequency, we must increase the string gauge by a factor \(d_2/d_1=\nu_1/\nu_2\). This is shown in Table 5. We can conclude from these numbers that:

• playing a set of 0.012 in E\(\flat\) standard will be slightly harder than playing a set of 0.011 in standard tuning;
• playing a set of 0.012 in D standard will be slightly easier than playing a set of 0.011 in standard tuning, but noticeably harder than 0.010.

Table 5: Rough equivalent string gauge when detuning. This table shows what string gauge in standard tuning would have a similar tension to a 0.012 in string set, when detuning each string by a half-step (E\(\flat\) standard) or a whole step (D standard).

String	Diameter	E\(\boldsymbol{\flat}\) standard	D standard
E (high)	d=0.012 in	\(d_{eq}\simeq0.0113\) in	\(d_{eq}\simeq0.0107\) in
B	d=0.016 in	\(d_{eq}\simeq0.0151\) in	\(d_{eq}\simeq0.0143\) in
G	d=0.023 in	\(d_{eq}\simeq0.0217\) in	\(d_{eq}\simeq0.0205\) in
D	d=0.031 in	\(d_{eq}\simeq0.0293\) in	\(d_{eq}\simeq0.0276\) in
A	d=0.041 in	\(d_{eq}\simeq0.0387\) in	\(d_{eq}\simeq0.0365\) in
E	d=0.051 in	\(d_{eq}\simeq0.0481\) in	\(d_{eq}\simeq0.0454\) in

We can also estimate the equivalent string gauge when changing the scale length. For instance, if we have a Telecaster (L=647.7 mm) with a set of 0.010, what would be the string gauge we would have to put on an ES 335 (L=628.7 mm) to keep the same tension. In this case, the string gauge ratio is simply inversely proportional to the scale length ratio 647.7/628.7=1.03. This is shown in Table 6. It is quite small. Playing with the same set of string on a Fender will require roughly 6% more strength than on a Gibson.

Table 6: Rough equivalent string gauge when going from a Fender scale length to a Gibson scale length.

String	Fender string gauge	Equivalent Gibson string gauge
E (high)	d=0.010 in	\(d_{eq}\simeq0.0103\) in
B	d=0.013 in	\(d_{eq}\simeq0.0134\) in
G	d=0.017 in	\(d_{eq}\simeq0.0175\) in
D	d=0.026 in	\(d_{eq}\simeq0.0268\) in
A	d=0.036 in	\(d_{eq}\simeq0.0371\) in
E	d=0.046 in	\(d_{eq}\simeq0.0474\) in

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