The common way of intonation adjustment for guitars / basses are expressed as follows:
"Compare pitches between the 12th fret harmonic and 12th fretted note using an electronic chromatic guitar tuner. Then, adjust the position of the corresponding bridge saddle."
This way is principally true in ideal physics; however, with this way I often feel difficulty of intonation adjustment of actual guitars which are not always perfectly manufactured.
Moreover, there are cases I feel musically happy in more even intonation among fretted notes in wide fret numbers, instead of pinpoint accuracy of only on 12th fret and open.
Thus, I propose another way to intonate a guitar, based on hearing two fretted notes, and without too much dependent on an expensive high-accuracy chromatic guitar tuner.
The way is for guitars with 6-way type bridges and in the standard tuning.
Assume the starting point is that except for intonation the guitar is already setup in at least a fairly good status particular with neck straightness and balanced string height.
1st string (3rd fret vs 15th fret)
Adjust the 1st string bridge saddle to align both of:
Set 5th string 12th fret fretted note (=A3) in one octave below 4th string 7 fret harmonic note (3rd harmonic =A4).
5th string 12th fret twice harmonic note (=A4, i.e., picking harmonics on 24 fret in 12th fretted) is better.
The difference of perfect 5th between true temperament (3/2 times) and equal temperament (2(7/12) = 1.49830707668... times) should be compromise. This difference can be more strictly tuned up but it probably makes no sense because of neck torsion during ordinal playing.
Set 6th string 12th fret fretted note (=E3) in one octave below 5th string 7 fret harmonic note (3rd harmonic =E4).
Repeat until complete alignment.
Repeat from 1st string until satisfied.
It is possible for 5th and 6th strings to apply adjustment using two fretted notes instead of 12th fret harmonics method:
5th string (2nd fret vs 14th fret)
Tune 5th string as 5th string 2nd fret fretted note (=B2) in precisely one octave below 2th string open note (=B3).
Adjust the 5th string bridge saddle to align 2nd string open note (=B3) and 5th string 14th fret fretted note (=B3)
Repeat until complete alignment.
6th string (3rd fret vs 15th fret)
Tune 6th string as 6th string 3rd fret fretted note (=G2) in precisely one octave below 3rd string open note (=G3).
Adjust the 6th string bridge saddle to align 3rd string open note (=G3) and 6th string 15th fret fretted note (=G3)
Repeat until complete alignment.
However, I wonder applying two fretted notes method for 5th and 6th strings brings not so important benefit in general use of majority of guitarists; because there are less opportunities to play very high fret notes on 5th or 6th strings. Moreover, thick core wire of 5th and 6th strings always bring slight shift of tuning to due to their stiffness.
Well, back to the original topic. For fine condition guitars, this method provides almost identical result compared to traditional 12th fret harmonics method.
Even in case of poor condition guitars, this method can keep slightly better intonation in average on fretted notes in wide range of fret numbers than traditional 12th fret harmonics method.
However, in case of poor condition guitars, the drawback of this method is slight enlargement of pitch deviation of open harmonics between the pairs of adjacent strings. This leads to two drawbacks:
Sonics of open chords might be somewhat sacrificed. Specifically, this method keeps sonics of open chords of D, A, and Am, barre chords on low frets, single notes from the midrange to the high frets better than traditional 12th fret harmonics method, while the open chords of C and G might be deteriorated. Power chords on low strings low frets don't change for better or worse.
It makes "harmonics tuning" inappropriate. Many guitarists tend to use harmonics tuning for quick tuning recover on stages; therefore, the guitarist should understand the characteristic of this method.
(Note that, the exception, the pair of harmonics on 2nd string 5th fret and 3rd string 4th fret is always different regardless of this method. This pair is not appropriate in common for accurate tuning without deep mathematical understanding of equal temperament.)
Anyway, I think that some compromises and choices are necessary depending on the song, playing style, and condition of the instrument.
This method is my original. I couldn't find similar examples just by looking for them (at least in Japanese) but that should be just due to my bad searching.
Is this common sense among guitar techs or repair persons? Or is it simply not written as a public document? I wonder what the truth is.
Miscellaneous and trivia
Why sometimes not so good on method comparing 12th fret fretted note with harmonic note
A high accuracy tuner is required. Not only equipped with strobe meter mode but also robust for picking noise. Moreover, in the view point of physics and mathematics theories, frequency domain resolution and time domain resolution cannot be satisfied in a same time.
Nonlinearity of scale due to poor guitar conditions:
Neck condition. Static irregularity (concave, convex, wrap), torsion by handling during played.
Fret irregularity. Construction accuracy problem in the factory, wearing over long usage.
Nut condition. Especially, irregularity of intonation between open and 1st fret due to too shallow string groove and/or misplaced nut.
Physics
Fretted note contains rich harmonics including odd ones, while 12th fret harmonic note does not contain any odd harmonics. Especially, the absence of 3rd harmonic, which is slightly higher than the perfect 5th on equal temperament, sounds 12th fret harmonic note slightly flat for ears accustomed to equal temperament compared to fretted note on 12th fret, even if accurate fundamental frequency. (to be confirmed by some specialist)
The vibration of string is generally divided into two dimensions: fret direction and body thickness direction. The two vibrations exist simultaneously from just after picking to the end of the note.
They cause a kind of 'twang' due to the difference of stiffness of neck and body of the guitar between fret direction and thickness direction. Thus, the string moves in torsion circular pattern. Fret direction vibration is damped on fretted notes but not damped in open note. Such dynamics are not easy to recognize with electronic tuners.
Worn frets might lead to raised intonation due to shortening the actual length of string.
True vibration length of string is very slightly shortened from physical string length due to stiffness of string itself nearby nut and bridge saddles. However, this proposed method does not consider this effect.
Musical encyclopedia for students, ONGAKU NO TOMO SHA corporation., 1978 (in Japanese)
History of this page
2023/03/04 first published together with Japanese page
2023/03/05 give tentative name "two fretted notes method" in contrast with traditional (12th fret harmonics) method; add some description for 5th and 6th strings